Question 1

If the letters are given repeating numerical values A = 4, B = 3, C = 2, D = 4, E = 3, F = 2 and so on in the same 4, 3, 2 pattern, what is the total of the numerical values of the letters in the word "SICK"?

Accepted Answer

Introduction / Context: This puzzle combines alphabet positions with a repeating numerical pattern. Instead of assigning each letter its usual place value (such as A = 1, B = 2, and so on), the question defines a special code where letters cycle through the values 4, 3, and 2. The task is to decode the given word SICK using this pattern and then add the numerical values of its letters. Questions like this test your attention to patterns, your ability to extend a sequence, and your skill in mapping letters to numbers correctly. Given Data / Assumptions: The mapping is defined as A = 4, B = 3, C = 2, D = 4, E = 3, F = 2. The pattern 4, 3, 2 repeats continuously through the alphabet. The word to be evaluated is SICK. We assume standard alphabet order: A is the 1st letter, B the 2nd, up to Z as the 26th. We need the sum of the coded values of S, I, C, and K. Concept / Approach: The core idea is to see that the code uses a repeating cycle of length three: 4, 3, 2, then again 4, 3, 2, and so on. That means letters whose positions differ by 3 share the same value. Once you know the position number of a letter in the alphabet, you can determine where it falls in the cycle by using remainders when divided by 3. After finding the individual values for S, I, C, and K, you simply add them to get the total asked in the question. Step-by-Step Solution: Step 1: Note the cycle: A (1) = 4, B (2) = 3, C (3) = 2, D (4) = 4, E (5) = 3, F (6) = 2, and so on. Step 2: Write the alphabet positions of the letters in SICK. S is the 19th letter, I is the 9th, C is the 3rd, and K is the 11th. Step 3: Because the cycle has length three, use the remainder when each position is divided by 3 to identify its value: remainder 1 maps to 4, remainder 2 maps to 3, and remainder 0 maps to 2. Step 4: For S (19), 19 divided by 3 gives remainder 1, so S has value 4. Step 5: For I (9), 9 divided by 3 has remainder 0, so I has value 2. Step 6: For C (3), 3 divided by 3 also has remainder 0, so C has value 2. Step 7: For K (11), 11 divided by 3 has remainder 2, so K has value 3. Step 8: Add all the values: 4 (S) + 2 (I) + 2 (C) + 3 (K) = 11. Verification / Alternative check: As an additional check, you can extend the explicit mapping a bit further. After F (6) = 2, G (7) = 4, H (8) = 3, I (9) = 2, J (10) = 4, K (11) = 3, and L (12) = 2. This confirms the values for I and K directly from the sequence. Doing so shows that S, which is the 19th letter, lands on the same pattern position as A, D, G, J, and M that all have value 4. This consistency confirms that the total of 11 is correct. Why Other Options Are Wrong: The totals 12, 10, and 9 arise if you misread the pattern, forget that it repeats every three letters, or assign simple alphabetical values instead of the coded ones. For example, using A = 1, B = 2 style coding or making a mistake on one letter value will shift the final sum to an incorrect option. Common Pitfalls: A common error is to assume the values are based on alphabetical position only or to think they change linearly. Another pitfall is to stop the pattern after F instead of repeating it, which leads to guessing for later letters. To avoid mistakes, always write out the cycle clearly and use division by 3 to place each letter correctly within that cycle. Careful calculation for each letter prevents small slips that could change the total. Final Answer: The total of the numerical values of the letters in the word SICK under the given code is 11.

Question 2

Two fathers and two sons sit down to eat eggs for breakfast. They eat exactly three eggs, and yet each person gets one whole egg. How is this possible?

Accepted Answer

Introduction / Context: This is a very popular family relationship riddle. At first, the numbers sound impossible: two fathers and two sons would normally suggest four people, so three eggs should not be enough for everyone to get one whole egg. The question is designed to test your understanding that the same person can belong to more than one relationship group at the same time. In particular, a person can be both a father and a son. This insight is the key to resolving the apparent contradiction and explaining the situation clearly. Given Data / Assumptions: The group is described as two fathers and two sons. Exactly three eggs are available and all of them are eaten. Each person receives one whole egg, not half or a shared piece. No eggs are wasted or partly eaten. The riddle uses everyday family relationships, not step parents or adoptive complications. Concept / Approach: The central concept is overlapping roles in a family. The phrase two fathers and two sons does not necessarily mean four distinct people. One person can be both a father and a son at the same time. For example, a man who has a father and also has a son fits both categories. The typical solution in this riddle is to imagine three generations present together: a grandfather, his son, and his grandson. Counting roles rather than people creates the apparent paradox that the riddle uses to confuse the reader at first. Step-by-Step Solution: Step 1: Assume there are three generations: an older man, his child, and that child own child. Step 2: Call the oldest man the grandfather, the middle person the father, and the youngest the son. Step 3: Identify the roles: the grandfather is a father but is not described as a son in this story. The father is both a father to the grandson and a son to the grandfather. The grandson is a son but not a father. Step 4: Count the fathers. The grandfather and the middle generation man are two fathers. Step 5: Count the sons. The middle generation man and the grandson are two sons. Step 6: Notice that there are two fathers and two sons, but only three individuals, because the middle man is counted twice in different roles. Step 7: Since there are three people, three eggs are enough for each person to receive one whole egg, which matches the description in the riddle. Verification / Alternative check: To verify, list the people as Person 1 (grandfather), Person 2 (father and son), and Person 3 (grandson). Person 1 and Person 2 are fathers, and Person 2 and Person 3 are sons. That satisfies the phrase two fathers and two sons. At the same time, the total head count is only three, so three eggs are exactly enough for each person to eat one egg. This matches every condition in the puzzle without any need for cutting, sharing, or skipping eggs. Why Other Options Are Wrong: The idea of cutting eggs into pieces contradicts the statement that each person had an egg, which strongly implies whole eggs. Saying one person does not eat an egg also conflicts with each person had an egg. Counting the cook as a father adds an extra person and does not reduce the total number of mouths to three. Only the explanation with three generations fits both the relationships and the numbers. Common Pitfalls: Many learners assume that two fathers and two sons always means four people, which is the main trap in the riddle. Others suspect tricks involving partial eggs or people refusing food, but such solutions are more complicated than necessary. Remember that in family puzzles, the same person can fill more than one relational role. Whenever you see a similar description with overlapping family labels, always check if a grandfather, father, and son trio can resolve the puzzle. Final Answer: The group consists of three people, a grandfather, his son, and his grandson. That way there are two fathers and two sons, but only three people, so three eggs are enough for each person to get one egg.

Question 3

On a standard clock face, you divide the numbers 1 to 12 into three regions using exactly two straight lines so that the sum of the numbers in each region is the same. What is this common sum for each region?

Accepted Answer

Introduction / Context: This is a classic clock and number puzzle. You imagine the face of a standard analog clock with numbers from 1 to 12 placed around a circle. The challenge is to divide these numbers into three groups using only two straight lines drawn across the clock. Each of the three regions created by the lines must have the same total sum of the numbers inside it. The question asks what that equal sum must be, not the exact placement of the lines. This tests your understanding of total sums and how they can be split equally. Given Data / Assumptions: The clock face has numbers 1 through 12, each used exactly once. Two straight lines divide the circular face into three regions. Each region contains some of the numbers from 1 to 12. The sum of the numbers in each region is required to be the same. We assume no number is cut in half; each number belongs to exactly one region. Concept / Approach: The key idea is that if three groups have equal totals, their total sum must be divisible by three. First, you calculate the sum of the numbers from 1 to 12. Once you know the total, you divide by three to get the required sum in each region. This step gives the answer directly, even before worrying about exactly how the lines are drawn. The geometrical arrangement can be figured out later, but for this multiple choice question, it is enough to work with arithmetic and divisibility. Step-by-Step Solution: Step 1: Compute the total of the numbers from 1 to 12. Step 2: Use the formula for the sum of the first n natural numbers or simply add: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12. Step 3: This sum is 78. You can check it quickly by pairing numbers: 1 + 12 = 13, 2 + 11 = 13, 3 + 10 = 13, 4 + 9 = 13, 5 + 8 = 13, and 6 + 7 = 13, giving six pairs of 13, so 6 * 13 = 78. Step 4: If the three regions have equal sums, then each region must contain 78 / 3 = 26 as its total. Step 5: Confirm that 26 is one of the options given in the problem. Step 6: Note that other totals such as 24, 28, or 30 would not produce a combined total of 78 when multiplied by 3, so they cannot be correct. Verification / Alternative check: To verify further, you may try to construct one actual grouping. One known solution is to group the numbers so that three sets each sum to 26. For example, one possible grouping is 12 + 11 + 1 + 2, another is 10 + 9 + 3 + 4, and the third is 8 + 7 + 6 + 5. Each group sums to 26 and can be separated by drawing two straight lines appropriately on the clock face. This proves that such a division is possible and confirms that 26 is indeed the correct common sum. Why Other Options Are Wrong: If each region summed to 24, the total would be 3 * 24 = 72, which is less than 78, so some numbers would be unaccounted for. A sum of 28 would give 84 in total, which is higher than 78 and impossible using only numbers 1 to 12. Similarly, a sum of 30 would require a total of 90, again impossible with the fixed set of numbers. Therefore 26 is the only value consistent with the total of 78. Common Pitfalls: A common mistake is to focus on drawing imaginary lines on the clock before working out the arithmetic. This can be confusing and time consuming. Another pitfall is assuming that the equal sum must be a round number like 24 or 30 and guessing based on that. The safe approach is always to start with the total of the numbers and then divide by the number of regions. Once that is done, any further geometric reasoning becomes much easier. Final Answer: The equal sum of the numbers in each of the three regions must be 26.

Question 4

Using exactly four 9s, one 1, and only a single arithmetic symbol, which of the following expressions makes a value of 100?

Accepted Answer

Introduction / Context: This is a digit and operation puzzle that asks you to reach 100 in a restricted way. You must use exactly four instances of the digit 9 and one instance of the digit 1, and you are allowed to use only one arithmetic symbol. That symbol might be plus, minus, multiply, or divide, but you cannot combine several different operators. The challenge is to arrange the digits into one or more multi digit numbers and then connect them with a single operator so that the final result is exactly 100. This tests creativity, number sense, and careful reading of constraints. Given Data / Assumptions: You must use the digits 9, 9, 9, 9, and 1 exactly once each. You may form multi digit numbers such as 99 or 199. You are allowed only one arithmetic symbol between numbers. The expression must evaluate exactly to 100. Standard arithmetic rules for addition, subtraction, multiplication, and division apply. Concept / Approach: The key idea is to think about how close you can get to 100 using multi digit numbers made from the given digits. Subtraction is particularly useful because it can turn a difference of two multi digit numbers into an exact target. If you arrange three of the digits as 199 and the remaining two as 99, then 199 minus 99 is exactly 100. This uses the digits 1, 9, 9, 9, and 9 and only a single subtraction sign, satisfying all the conditions laid out in the puzzle. Step-by-Step Solution: Step 1: Try to form numbers that are reasonably close to 100 so that a simple operation can reach 100. Step 2: Consider forming 199 from digits 1, 9, and 9, leaving 9 and 9 to form 99. Step 3: Test the expression 199 - 99. Step 4: Calculate 199 - 99. The difference is 100. Step 5: Check the constraints: the digits used are exactly four 9s and one 1, and only one subtraction symbol appears. Step 6: Verify that the other answer options either use more than one operation symbol or do not give exactly 100. Verification / Alternative check: For a quick verification, add 99 and 100 to see that their sum is 199. This confirms that 199 is exactly 100 more than 99, so 199 minus 99 must be 100. It is also helpful to check that you did not accidentally introduce extra digits or operations. In this expression, the multi digit numbers 199 and 99 are built strictly from the four 9s and one 1. No extra symbol beyond the single minus sign appears, so the solution is fully valid under the puzzle rules. Why Other Options Are Wrong: The expression 91 + 9/9 uses both addition and division, so it does not respect the one symbol rule. The expression 9 * 9 + 1 + 1 uses multiplication and two additions, again violating the constraint on the number of symbols. The expression 99 + 1/9 uses both addition and division and also does not evaluate to 100. Thus, even if some of these get numerically close to 100, they do not follow the strict requirement of using only one arithmetic symbol. Common Pitfalls: A typical mistake is to overlook the restriction on the number of operations and to focus only on reaching 100. Many clever looking expressions that equal 100 use multiple operations or parentheses, which are not allowed here. Another pitfall is to think only in terms of single digit numbers and forget that multi digit numbers like 99 or 199 are permitted. Once you remember that 199 and 99 are possible and consider subtraction, the path to the correct answer becomes straightforward. Final Answer: The correct expression is 199 - 99, which uses four 9s, one 1, and exactly one arithmetic symbol to make 100.

Question 5

Which whole number, when multiplied by itself (that is, squared), gives the palindromic number 12345678987654321?

Accepted Answer

Introduction / Context: This puzzle asks you to identify a number whose square is the striking palindromic value 12345678987654321. A palindromic number reads the same forwards and backwards, and this particular one increases from 1 up to 9 and then decreases back to 1. The question wants the whole number which, when multiplied by itself, produces this pattern. Such problems test your familiarity with special numerical patterns and your ability to reason about the approximate size of square roots and how they relate to the length of the resulting number. Given Data / Assumptions: The target number is 12345678987654321. We are looking for a whole number n such that n * n equals this target. The answer choices are 11111111, 111111111, 1111111111, and 123456789. We assume normal rules of decimal multiplication. The structure of the target number suggests a pattern rather than a random square. Concept / Approach: The key idea is that repeating numbers like 11, 111, or 111111111 often produce attractive patterns when squared. For example, 11 squared is 121, and 111 squared is 12321. These squares show a growing sequence of digits and then a decrease. Extending that idea, 111111111 squared gives a much longer palindromic number, and in fact, it yields exactly 12345678987654321. Knowing this pattern or at least recognising that a nine digit string of ones is the right size helps to select the correct option without doing full long multiplication in the exam hall. Step-by-Step Solution: Step 1: Estimate the number of digits in the square root. The target has 17 digits, so the square root should have 9 digits because a 9 digit number squared typically has 17 or 18 digits. Step 2: Among the options, 111111111 is the only 9 digit number. Step 3: Recall related patterns: 11 * 11 = 121 and 111 * 111 = 12321. These show the same building then reversing structure. Step 4: Extend the pattern mentally: 111111111 squared is known to create 12345678987654321, matching both the length and the digit pattern of the target. Step 5: Rule out 11111111 (8 digits) and 1111111111 (10 digits), because their squares would not have exactly 17 digits. Step 6: Rule out 123456789, because its square would be larger and would not follow the symmetric pattern of digits that the target has. Verification / Alternative check: If you wish to check further, note the general pattern: if you square a number made up of n ones, you get a palindrome that climbs from 1 up to n and back down, when n is 1 through 9. For n = 9, the square of 111111111 is 12345678987654321. Although doing full long multiplication is possible, recognising this well known pattern is faster and is often taught in number puzzle books. Additionally, the digit count argument reinforces that the correct root must have 9 digits, which uniquely points to 111111111 among the choices. Why Other Options Are Wrong: 11111111 has only 8 digits, so its square would have about 15 or 16 digits, not 17. 1111111111 has 10 digits, so its square would have 19 or 20 digits, longer than the target. The number 123456789, while famous in its own right, does not have the simple all ones structure that produces symmetric palindromes when squared, and its square does not equal the given number. Thus, they cannot be correct roots for 12345678987654321. Common Pitfalls: A frequent mistake is to guess based on the familiarity of 123456789 without considering the number of digits in the square root. Another pitfall is to underestimate how useful pattern recognition is in contest style questions and attempt heavy calculations under time pressure. In puzzles involving palindromic numbers with neat sequences, it is often best to think about special forms such as strings of ones or repeated digits before resorting to brute force arithmetic. Final Answer: The number whose square is 12345678987654321 is 111111111.

Question 6

The letters of the word "FAINTS" are rearranged in alphabetical order. How many of the letters occupy the same position in the original word and in the alphabetically arranged word?

Accepted Answer

Introduction / Context: This question is a word and position puzzle. You are given the word FAINTS and asked to compare its original arrangement with a new arrangement where the letters are sorted into alphabetical order. The goal is to count how many letters stay in exactly the same position in both versions. Problems of this type test your attention to detail and your ability to work with ordering of letters both visually and conceptually, which can appear in reasoning and verbal ability sections of competitive exams. Given Data / Assumptions: The original word is FAINTS. We must arrange its letters in alphabetical order based on standard English alphabetical sequence. The new ordered form uses the same six letters: A, F, I, N, S, and T. We then compare positions one by one between the original word and the alphabetical version. We count how many letters are in the same position in both words. Concept / Approach: The key idea is to perform an alphabetical sort of the letters and then match positions. Alphabetical order for the letters involved is A, F, I, N, S, T. Once you write the sorted version, you can line it up with the original word and check each position. You are not checking whether the same letters appear, but whether the same letter appears in the same position number in both sequences. That is what the question means by their order in the word and in the alphabetical order remaining the same. Step-by-Step Solution: Step 1: Write the original word with positions: position 1 F, position 2 A, position 3 I, position 4 N, position 5 T, position 6 S. Step 2: List the letters of FAINTS and sort them alphabetically: A, F, I, N, S, T. Step 3: Write the alphabetically ordered form with positions: position 1 A, position 2 F, position 3 I, position 4 N, position 5 S, position 6 T. Step 4: Compare the letters at each position in the two forms. Step 5: At position 1, original has F and sorted has A, which do not match. Step 6: At position 2, original has A and sorted has F, which do not match. Step 7: At position 3, original and sorted both have I, which is a match. Step 8: At position 4, original and sorted both have N, which is another match. Step 9: At position 5, original has T and sorted has S, which do not match. Step 10: At position 6, original has S and sorted has T, which do not match. Verification / Alternative check: From the step by step comparison, we have matches only at positions 3 and 4. That means exactly two letters, I and N, retain their positions when the word is sorted alphabetically. To confirm, you can quickly reconstruct: original FAINTS and sorted AFINST. Only the middle part IN appears in the same positions in both words. No other positions match. This double check ensures that no letter has been overlooked and that the count of two is correct. Why Other Options Are Wrong: One would be correct only if a single position matched, which is not the case. Three would require an additional matching position such as the first or last letter, but we see that F and A swap their places and S and T swap theirs when sorted. Nil would mean no position matches at all, which is clearly not true because both the third and fourth positions match. Therefore, Two is the only option consistent with our detailed comparison. Common Pitfalls: A common mistake is to misread the question and simply count how many letters are common between the two words, which will always be all of them because they are anagrams. Another pitfall is to arrange the letters incorrectly in alphabetical order, for example by misplacing F and A. To avoid errors, always write positions explicitly and compare them line by line rather than trying to do it mentally in one step. Final Answer: Exactly two letters, I and N, remain in the same position when FAINTS is rearranged alphabetically.

Question 7

You are going to visit your grandmother on Mother Day and want to give her two cakes. To reach her house you must cross seven bridges. At each bridge, a troll takes half of all your cakes but then kindly gives you back one cake. If you want to arriv…

Accepted Answer

Introduction / Context: This puzzle is a fun combination of algebra and story telling. You are carrying cakes and must cross seven bridges, and at each bridge a troll takes half your cakes but then gives one cake back. Despite these repeated losses and gifts, you want to end up with exactly two cakes when you reach your grandmother. The question asks how many cakes you must start with. Puzzles like this illustrate how working backwards can simplify what looks like a complicated sequence of operations. Given Data / Assumptions: There are exactly seven bridges to cross. At each bridge, the troll first takes half of your current cakes. After taking half, the troll always returns one cake to you. The process is identical at every bridge. You want to arrive at your grandmother house with exactly two cakes. Concept / Approach: The key concept is reverse working. Instead of following the story forward from the start, start at the end with the known final number of cakes and move backward bridge by bridge. At each step in reverse, you invert the operations of the troll. In the forward direction the troll takes half and then gives one, so in the reverse direction you first remove the gift of one and then double the remaining cakes. By repeating this reverse operation seven times, you can find the initial number of cakes you must carry from home. Step-by-Step Solution: Step 1: Let the number of cakes you have just after crossing the last bridge be 2, since you want to arrive with two cakes. Step 2: Reverse the troll action. In reverse, before the troll gave you 1 cake, you must have had 2 minus 1 equal to 1 cake. Step 3: Before the troll took half of your cakes, you must have had double that amount. So before the last bridge event, you had 1 * 2 = 2 cakes. Step 4: Notice that this reverse calculation took you from after bridge 7 back to before bridge 7, and the number is still 2. Step 5: Repeat the same reverse logic for bridge 6. After bridge 6 you would have 2 cakes. Before the gift, that is 1 cake, and before the halving, that is again 2 cakes. Step 6: Continue this pattern backward across all seven bridges. Each time you reverse the operations, you end up with 2 cakes before the bridge. Step 7: After stepping back through all seven bridges in this way, you find that at the very start of the journey you still need only 2 cakes. Verification / Alternative check: To verify, run the story forward using 2 cakes as the starting number. Before bridge 1 you have 2 cakes. The troll takes half (1), leaving 1, and then gives 1 back, returning you to 2. You cross bridge 2 with 2 cakes and the same thing happens. This repeats at each of the seven bridges. After every bridge you still have 2 cakes. At the end of the seventh bridge, you arrive at your grandmother house with exactly two cakes, which confirms that starting with 2 works perfectly. Why Other Options Are Wrong: If you start with 4, then at the first bridge half is taken (2), one is returned (3), and now you carry 3 cakes into the next bridge. Repeating the process will not conveniently bring you back to exactly 2 at the end. Similar mismatches occur for starting values of 6 or 8. Only 2 remains stable under the operation of halving and then adding 1 through multiple repetitions, which is why the backward reasoning leads you directly to that value. Common Pitfalls: A common mistake is to try to simulate the process forward with an unknown starting number, which can become confusing. Algebraic equations can be used, but they often seem heavy for a story puzzle. Working backwards is cleaner and more intuitive in this context. Another pitfall is to misinterpret half of the cakes if the number becomes odd. In this puzzle, starting with 2 avoids such issues because you always remain at 2 after each operation, making the scenario consistent and easy to track. Final Answer: You must start the journey with 2 cakes in order to arrive with exactly two cakes after crossing all seven bridges.

Question 8

A wooden block has dimensions 6 cm by 4 cm by 1 cm. The two faces measuring 6 cm x 4 cm are painted green, the two faces measuring 6 cm x 1 cm are painted red, and the two faces measuring 4 cm x 1 cm are painted black. The block is cut into 24 unit …

Accepted Answer

Introduction / Context: This is a three dimensional geometry and counting puzzle. You start with a rectangular wooden block whose faces are painted in different colours: two opposite large faces are green, two long thin faces are red, and two short thin faces are black. The block is then cut into 1 cm cubes. The question asks how many of these small cubes show green paint on exactly one face and have no paint on the remaining five faces. Problems like this test spatial imagination and careful reasoning about which smaller cubes lie on edges, corners, or strictly inside larger painted faces. Given Data / Assumptions: The block measures 6 cm in length, 4 cm in breadth, and 1 cm in height. The two 6 cm x 4 cm faces are painted green. The two 6 cm x 1 cm faces are painted red. The two 4 cm x 1 cm faces are painted black. The block is cut into 1 cm cubes, so there are 6 * 4 * 1 = 24 small cubes. We want cubes that have exactly one green face and all other five faces unpainted. Concept / Approach: The key idea is to understand which positions in the block give cubes one green face but keep them away from the edges and corners. Every cube that lies on a green face will have at least one green face. However, cubes at the corners or along the edges where green meets red or black will have extra painted faces in red or black. Only those cubes that are on a green face, but not on any edge of that face, will have exactly one face painted green and no other colour on the remaining faces. Because the height is only 1 cm, the block has a single layer of cubes between the two green faces. Step-by-Step Solution: Step 1: Visualise the block as a 6 by 4 grid of 1 cm cubes on the top green face and another 6 by 4 grid on the bottom green face. Step 2: Focus first on the top green face. All 24 cubes belong to either the top face or the bottom face, since the height is 1 cm. Step 3: On the top green face, corner cubes have three painted faces (green plus one red plus one black) and edge cubes have two painted faces (green plus either red or black). Step 4: The only cubes with exactly one green face on the top layer are the interior cubes of the 6 by 4 rectangle, which do not touch any red or black sides. Step 5: Count those interior cubes on the top face by excluding the outer border. Along the length, exclude the two end columns, leaving 6 - 2 = 4 interior columns. Along the breadth, exclude the two border rows, leaving 4 - 2 = 2 interior rows. Step 6: The interior count on the top face is 4 * 2 = 8 cubes, each having exactly one green face and no other paint. Step 7: Repeat the same reasoning for the bottom green face. It has the same pattern, giving another set of 8 interior cubes with exactly one green face. Step 8: Add the two sets: 8 from the top face plus 8 from the bottom face gives 16 cubes. Verification / Alternative check: As a check, consider the total structure. There are 24 cubes. Corners and edges on the top and bottom faces account for those cubes that touch red or black faces. Specifically, on each green face there are 4 corner cubes and 8 edge cubes that are not corners, making 12 boundary cubes per face. Because the block is only one cube high, each physical corner or edge cube on the top has the same cube as boundary on the bottom. Counting carefully, all boundary cubes have more than one painted face, and the remaining 16 interior positions across the two green faces are the only ones that have exactly one green face and no other paint. Why Other Options Are Wrong: An answer of 8 would count only one green face, forgetting that there are two such faces. An answer of 24 would assume that every cube with green paint has only one green face, which ignores edges and corners where other colours appear. An answer of 4 is too small and would correspond only to a subset of interior cubes, not the full count from both faces. Only 16 matches the full interior positioning on both green faces. Common Pitfalls: A common error is to think only about the top green face and then stop, forgetting that the bottom face also contributes an equal number of cubes. Another pitfall is to miscount cubes on edges and corners, accidentally including cubes that have additional red or black paint. Drawing a simple 6 by 4 grid and marking border and interior positions is a reliable way to avoid miscounting and to see clearly which cubes qualify. Final Answer: There are 16 small cubes that have green paint on exactly one face and no paint on their other five faces.

Question 9

There are eight identical looking balls, but one of them is heavier than the others. Using a balance scale only twice, which strategy will always allow you to find the overweight ball?

Accepted Answer

Introduction / Context: This classic reasoning puzzle involves eight identical looking balls where exactly one is heavier than the others. You are given a balance scale, which can compare the weights of two groups of balls, and you are allowed to use the scale only twice. The challenge is to find a strategy that guarantees you will identify the heavier ball in these two weighings. This type of problem is common in aptitude tests and measures logical planning and optimal use of limited information. Given Data / Assumptions: There are 8 balls that look the same. Exactly one ball is heavier than the rest; all others have equal, lighter weight. You have a balance scale that indicates which side is heavier or whether the two sides balance. You may use the scale at most two times. You must always be able to identify the heavier ball, regardless of which one it is. Concept / Approach: The key concept is to divide the balls into groups so that each weighing gives maximum useful information. Because one ball is heavier, the first weighing should split the set in a way that either isolates the heavy ball in a smaller suspect group or shows that it lies outside the weighed balls. Dividing the balls into groups of 3, 3, and 2 allows you to use the first weighing to decide whether the heavy ball is among 6 balls on the scale or among 2 balls off the scale. The second weighing then narrows down the suspect group to a single ball. Step-by-Step Solution: Step 1: Label the balls conceptually as A, B, C, D, E, F, G, and H. Step 2: For the first weighing, place A, B, and C on the left pan and D, E, and F on the right pan. Step 3: Observe the result of the first weighing. There are two possibilities: the scale balances, or one side is heavier. Step 4: If the scale balances, then all six balls A through F are of normal weight, and the heavier ball must be among G and H. Step 5: For the second weighing in this balanced case, compare G against H. The heavier side contains the overweight ball. Step 6: If, in the first weighing, one side is heavier (for example, left side heavier with A, B, C), then the heavy ball must be among those three balls because the other three on the lighter side must be normal. Step 7: For the second weighing in this unbalanced case, pick any two balls from the heavier group, say A and B, and compare A against B. Step 8: If A and B balance, then C is the heavier ball. If one of them is heavier, that ball is the overweight one. Verification / Alternative check: To verify the strategy, examine all possible locations of the heavy ball. If it is G or H, the first weighing will balance, and the second weighing between G and H will identify it. If it is A, B, C, D, E, or F, then the first weighing will tilt towards the side containing that ball. By limiting the suspect group to three and then doing a simple comparison between two of them, the second weighing always isolates the heavy one. In every scenario, you have an unconditional method to find the overweight ball in no more than two weighings. Why Other Options Are Wrong: Weighing 4 balls against 4 balls as in option A can tell you which group contains the heavy ball, but then you still have four candidates and only one weighing left, which is not enough to guarantee a unique identification. Option C with 2 against 2 then 2 against 2 splits information poorly and does not necessarily isolate a single ball in two weighings. Option D is not a correct use of a balance scale and would require far more than two weighings or would not give any clear comparison at all. Common Pitfalls: A common mistake is to divide the balls into two equal groups of four, which feels natural but does not optimise information for a second weighing. Another pitfall is to ignore the fact that the scale gives a trinary outcome (left heavy, right heavy, or balance) and to treat it as if it only gave one bit of information. Recognising that a 3, 3, and 2 split makes full use of the scale outcomes is essential. By planning carefully before touching the scale, you ensure that every possible result leads you closer to the solution. Final Answer: The correct strategy is: First weigh 3 balls against 3 balls. If they balance, weigh the remaining 2 against each other; if not, weigh 1 against 1 from the heavier group of 3.

Question 10

In this number riddle, how can you use eight instances of the digit 8, with only addition, to make a total of 1,000?

Accepted Answer

Introduction / Context: This is a popular digit puzzle about constructing 1,000 using only the digit 8 exactly eight times and using only addition as the operation. The challenge is to decide how to group the eights into multi digit numbers such as 88 or 888 so that their sum is exactly 1,000. This tests creativity, your ability to see patterns in numbers, and your skill in working within tight constraints where multiplication or subtraction are not allowed. Given Data / Assumptions: You must use the digit 8 exactly eight times. You are allowed to use only addition operations. You may form multi digit numbers like 88 or 888 as long as you still use only the digit 8. The final sum must be exactly 1,000. The expression must be a straightforward sum with no other symbols. Concept / Approach: The main idea is to think of 1,000 as slightly less than 1,000 from a large number like 888 and then fill the remaining gap with additional numbers built from the digit 8. If you begin with 888, you need 112 more to reach 1,000. That extra 112 can be formed by 88 + 8 + 8 + 8, which together use five eights. Combined with the three eights in 888, that makes exactly eight eights. The sum 888 + 88 + 8 + 8 + 8 equals 1,000, satisfies the digit count, and uses addition only. Step-by-Step Solution: Step 1: Start with a large number made from eights, such as 888, which uses three eights and is close to 1,000. Step 2: Compute the difference between 1,000 and 888. The gap is 1,000 minus 888, which equals 112. Step 3: Try to form 112 using only the digit 8 and addition. One convenient choice is 88 + 8 + 8 + 8. Step 4: Check that 88 + 8 + 8 + 8 equals 112. First, 88 + 8 = 96, then 96 + 8 = 104, and 104 + 8 = 112. Step 5: Combine the expressions: 888 + 88 + 8 + 8 + 8. Step 6: Add them step by step: 888 + 88 = 976, 976 + 8 = 984, 984 + 8 = 992, 992 + 8 = 1,000. Step 7: Count the eights: 888 uses three digits 8, 88 uses two, and each of the three single 8s adds one more, for a total of 3 + 2 + 1 + 1 + 1 = 8 eights. Verification / Alternative check: To verify, confirm both the arithmetic and the digit count. The sum clearly equals 1,000 based on straightforward addition. The eights appear in groups: 888 has three, 88 has two, and each solitary 8 has one. This totals eight eights exactly. There are no other digits present and only addition signs are used. Comparing with typical solutions found in puzzle books shows that 888 + 88 + 8 + 8 + 8 is the standard representation used to answer this riddle, which supports its correctness. Why Other Options Are Wrong: The expression with eight copies of 88 added together uses sixteen eights in total and sums to 704, which is far below 1,000. The expression 888 + 88 + 11 + 13 introduces digits 1 and 3, which violates the rule that only 8 may be used. The expression 888 + 8 + 8 + 8 + 8 + 8 + 8 + 8, although it uses eight eights, sums to 888 + 56 = 944, not 1,000. Therefore, only the option 888 + 88 + 8 + 8 + 8 meets all conditions. Common Pitfalls: A common mistake is to use too many eights or to inadvertently use other digits, especially when trying to form numbers like 80 or 20. Another pitfall is to allow yourself to use multiplication, such as 8 * 125, which is not permitted here. To avoid these issues, always check that only the digit 8 appears in the numbers, that there are exactly eight of them, and that plus signs are the only operations used. Building the sum around 888 is a simple and reliable strategy that satisfies all requirements. Final Answer: One correct way is 888 + 88 + 8 + 8 + 8, which uses eight eights and only addition to make 1,000.

Question 11

A group of cats got together and decided to kill, between them, a total of 999919 rats. Every cat killed the same number of rats, and each cat killed more rats than the total number of cats. How many rats did each cat kill?

Accepted Answer

Introduction / Context: This puzzle question tests your understanding of basic number theory, factors of an integer, and simple logical reasoning. We are told that a certain number of cats together killed exactly 999919 rats, that each cat killed the same number of rats, and that each cat killed more rats than there were cats in the group. The task is to determine how many rats each cat killed from the given options. Given Data / Assumptions: Total number of rats killed altogether = 999919. There are n cats, where n is a positive integer. Each cat kills exactly k rats, where k is a positive integer. All cats kill the same number of rats, so total rats = n * k. Each cat kills more rats than the total number of cats, so k > n. We assume the given answer choices contain the correct value of k. Concept / Approach: The key idea is to treat this as a factorisation puzzle. Since n * k = 999919, both n and k must be positive integer factors of 999919. The condition that each cat kills more rats than there are cats (k > n) will help us decide which factor is k and which is n. We can use the options as possible values of k, and check which one divides 999919 exactly and also satisfies k > n. Step-by-Step Solution: Step 1: Let the number of cats be n and the number of rats killed by each cat be k.Step 2: From the statement, we have n * k = 999919.Step 3: Test the answer options as possible values of k: 991, 1000, 1009 and 2000.Step 4: Check divisibility: 999919 / 1000 is not an integer and 999919 / 2000 is not an integer, so 1000 and 2000 cannot be k.Step 5: Compute 999919 / 991 = 1009 (an integer), and 999919 / 1009 = 991 (also an integer). So the factor pairs are (n, k) = (991, 1009) or (1009, 991).Step 6: Apply the condition k > n. Only the pair n = 991, k = 1009 satisfies k > n, so each cat must have killed 1009 rats. Verification / Alternative check: You can quickly verify the answer by multiplying 991 * 1009. Using arithmetic or a calculator, 991 * 1009 = 999919, which matches the total number of rats given in the problem. Also check the inequality: 1009 > 991, so each cat killed more rats than the number of cats. This confirms that k = 1009 is consistent with all conditions. Why Other Options Are Wrong: Option 991 is wrong because if k = 991, then n = 1009 and the condition k > n would be violated. Option 1000 and option 2000 are wrong because 999919 is not divisible evenly by 1000 or 2000, so they cannot be the exact number of rats killed by each cat while still giving a whole number of cats. Common Pitfalls: Students sometimes forget to use the condition that each cat kills more rats than there are cats, and they may simply pick any factor that appears reasonable. Another common error is to assume that the largest factor in the options must be the answer without checking both divisibility and the inequality k > n. Careful reading of the statement and systematic checking of the factor pairs helps avoid these mistakes. Final Answer: The correct number of rats killed by each cat is 1009.

Question 12

Jim, Jeanne, Louis and Anne want to cross a bridge at night with only one torch. Only two people can cross at a time, they must walk together at the speed of the slower person, and their individual crossing times are 1, 2, 5 and 10 minutes respectiv…

Accepted Answer

Introduction / Context: This classic bridge and torch puzzle tests logical reasoning, optimisation, and the ability to work with time constraints. Four people with different walking speeds must cross a bridge at night using a single torch, with only two allowed on the bridge at the same time. The challenge is to find a crossing sequence that allows everyone to cross in no more than 17 minutes, given their individual times of 1, 2, 5 and 10 minutes. Given Data / Assumptions: There are four people: Jim (1 minute), Jeanne (2 minutes), Louis (5 minutes) and Anne (10 minutes). Only two people can be on the bridge at the same time. They have only one torch, and it must always be carried by someone crossing. When two people cross together, the time taken is equal to the slower person's time. The goal is to get everyone across within a total of 17 minutes. Concept / Approach: The key to this type of puzzle is to minimise the time spent by the slow walkers on the bridge, while using the fastest people to shuttle the torch back and forth efficiently. We must design a sequence of crossings so that the two slowest individuals (Louis and Anne) cross together only once, and the fastest individuals (Jim and Jeanne) handle most of the returns with the torch. Step-by-Step Solution: Step 1: First, Jim (1 minute) and Jeanne (2 minutes) cross together. Time used: 2 minutes.Step 2: Jim, the fastest, returns with the torch. Time used: 2 + 1 = 3 minutes total.Step 3: Next, Louis (5 minutes) and Anne (10 minutes), the two slowest, cross together. They move at Anne's speed, so this takes 10 minutes. Time used: 3 + 10 = 13 minutes total.Step 4: Jeanne (2 minutes), who is already on the far side, now returns with the torch to bring back the torch for the final crossing. Time used: 13 + 2 = 15 minutes total.Step 5: Finally, Jim (1 minute) and Jeanne (2 minutes) cross together again, taking 2 minutes. Time used: 15 + 2 = 17 minutes in total. Verification / Alternative check: We can verify the timing by summing all segments: 2 (first crossing) + 1 (return) + 10 (slow pair crossing) + 2 (return) + 2 (final crossing) = 17 minutes. All four individuals have crossed to the far side and the total time does not exceed the required limit. Any alternative sequence that sends the slow walkers separately or uses them for returns will increase the total time beyond 17 minutes. Why Other Options Are Wrong: Option B makes Anne cross multiple times or involves inefficient pairings, leading to a total time greater than 17 minutes. Option C and option D similarly pair slow walkers in suboptimal ways or cause extra long returns, pushing the total over 17 minutes. Only option A organises the crossings in the most efficient manner. Common Pitfalls: Common mistakes include sending the slowest person back with the torch, splitting the two slowest people instead of having them cross together once, or not carefully adding the total travel times. Many learners also focus on minimising individual steps rather than the overall total. Thinking strategically about who should return and when is crucial to solving this puzzle correctly. Final Answer: The correct strategy is described in option A, which yields a total crossing time of exactly 17 minutes.

Question 13

In a certain country it is claimed that 1/2 of 5 equals 3. Assuming the same unusual rule of calculation is followed there, what value should 1/3 of 10 have in that country?

Accepted Answer

Introduction / Context: This puzzle question looks like a fraction problem, but it is really testing your ability to spot an unconventional rule hidden behind a simple statement. In a certain country, it is given that 1/2 of 5 equals 3, which clearly does not match standard arithmetic. The question asks you to apply the same rule to find the value of 1/3 of 10. You therefore need to infer the special rule from the first statement and then apply it consistently to the second. Given Data / Assumptions: In the puzzle setting, 1/2 of 5 is defined to be 3. We assume the same rule is used for all similar fractional expressions in that country. We are asked to compute 1/3 of 10 using that same rule. The answer must be chosen from the options 4, 6, 8 and 10. We treat this as a logical pattern puzzle, not as standard school arithmetic. Concept / Approach: If 1/2 of 5 were computed in the usual way, it would be 5 / 2 = 2.5, not 3. The result 3 is close to 2.5, suggesting a rounding rule. A natural interpretation is that, in this imaginary country, whenever a fractional result is not a whole number, it is rounded up to the nearest integer (that is, they always round up). This process is also called rounding toward the next larger integer or taking the ceiling of the value. Step-by-Step Solution: Step 1: Interpret the rule from the given statement: 1/2 of 5 means 5 / 2 = 2.5 in ordinary arithmetic.Step 2: To match the puzzle's statement that 1/2 of 5 equals 3, the country must round 2.5 up to 3.Step 3: Assume that the same rounding-up rule is applied to other similar expressions.Step 4: Now compute 1/3 of 10 in ordinary arithmetic: 10 / 3 ≈ 3.3333.Step 5: Apply the same rule and round 3.3333 up to the next whole number, which is 4. Verification / Alternative check: To verify that this interpretation is consistent, you can check that it works with the original statement. Using ordinary division, 5 / 2 = 2.5 and 10 / 3 ≈ 3.3333. Rounding both results up gives 3 and 4 respectively, which matches the given special value and the answer we selected. No other simple and consistent rule explains both values in such a straightforward way. Why Other Options Are Wrong: Option 6 would require 10 / 3 to be interpreted as something like 5.8 rounded up, which does not match the basic proportion given. Option 8 and option 10 are even further away from 3.3333 and cannot be justified by the same rule that turns 2.5 into 3. Therefore these options do not preserve the consistent rounding pattern that the puzzle describes. Common Pitfalls: Many learners try to force a direct proportionality between 5 and 3, and then apply it algebraically to 10, which leads to complicated or inconsistent formulas. Others insist on using ordinary arithmetic without adjusting for the puzzle's special rule. The key is to recognise that this is an imaginary rule-based setting and to identify a simple rule that matches the original statement. Rounding up non-integer results is the simplest and most natural explanation. Final Answer: According to the special rule in that country, 1/3 of 10 should be 4.

Question 14

In the puzzle sequence 1, 11, 21, 1211, 111221, what is the next term that should come after 111221?

Accepted Answer

Introduction / Context: This question tests your ability to recognise and extend a famous pattern called the "look-and-say" sequence. Unlike simple arithmetic progressions or geometric progressions, here each term is generated by verbally describing the digits of the previous term. Understanding how to read off the previous number and convert that description back into digits is the key skill needed to find the next term after 111221. Given Data / Assumptions: The given sequence of terms is: 1, 11, 21, 1211, 111221. We are asked to determine the next term that follows 111221. Each term is formed from the previous term by "looking" at groups of identical digits and then "saying" how many of each digit there are. The answer must be selected from the provided options. We assume the pattern has no hidden arithmetic; it is purely descriptive. Concept / Approach: In the look-and-say sequence, you create the next term by reading the previous term digit by digit, counting consecutive identical digits, and then writing down the count followed by the digit. For example, the term 21 is read as "one 2, one 1", which becomes 1211. The same rule is applied repeatedly. Our task is to apply this description rule to the term 111221 in order to generate the next term. Step-by-Step Solution: Step 1: Write down the last given term: 111221.Step 2: Group consecutive identical digits: 111, 22, 1.Step 3: Read each group out loud: "three 1s", "two 2s", "one 1".Step 4: Convert this spoken description back into digits by writing the count followed by the digit: "three 1s" becomes 31, "two 2s" becomes 22, and "one 1" becomes 11.Step 5: Concatenate these parts: 31 + 22 + 11 = 312211. Thus, the next term is 312211. Verification / Alternative check: You can quickly verify the rule by applying it to earlier terms. Starting from 1, "one 1" becomes 11. From 11, "two 1s" becomes 21. From 21, "one 2, one 1" becomes 1211. From 1211, "one 1, one 2, two 1s" becomes 111221. This confirms that the same consistent rule has been used, so applying it to 111221 to get 312211 is correct. Why Other Options Are Wrong: Option 132112 does not correctly encode the groups in 111221, since it suggests different counts and ordering. Option 111222 treats the sequence as simply increasing digits rather than describing the previous term. Option 213211 also mixes counts and digits incorrectly. Only option 312211 faithfully represents "three 1s, two 2s, one 1". Common Pitfalls: Learners sometimes misread the groups, for example interpreting 111221 as "two 1s, one 1, two 2s, one 1" rather than grouping identical digits consecutively as 111, 22, 1. Another common error is to try to find an arithmetic pattern in the values themselves rather than seeing that the sequence is descriptive. Carefully grouping digits and strictly following the count-then-digit rule avoids these mistakes. Final Answer: The correct next term in the sequence is 312211.

Question 15

Consider a five-digit number such that, if the digit 1 is placed at the beginning of the number, the resulting six-digit number is exactly one third of the number formed by placing the digit 1 at the end. What is this original five-digit number?

Accepted Answer

Introduction / Context: This puzzle involves an unknown five-digit number that behaves in an interesting way when the digit 1 is placed either at the front or at the back. When 1 is placed at the beginning, the resulting six-digit number is exactly one third of the number obtained by placing 1 at the end. The question checks your ability to translate a word puzzle into an algebraic equation, manipulate place values and solve for the unknown number. Given Data / Assumptions: Let the original five-digit number be denoted by N. Placing 1 at the beginning yields a six-digit number: 1N (one followed by the digits of N). Placing 1 at the end yields another six-digit number: N1 (the digits of N followed by one). The number 1N is exactly one third of N1. The answer must be a five-digit number chosen from the options: 51428, 42857, 57142 and 42168. Concept / Approach: To solve this puzzle systematically, we translate the words into equations involving place values. If N is a five-digit number, then putting 1 at the beginning is equivalent to 100000 + N, because we place 1 in the hundred-thousands position. Placing 1 at the end is equivalent to 10 * N + 1, because we shift N one place to the left and append 1. The condition that the first result is one third of the second gives us a linear equation relating N and these expressions. Step-by-Step Solution: Step 1: Let N be the original five-digit number.Step 2: Number with 1 at the front: 100000 + N.Step 3: Number with 1 at the end: 10 * N + 1.Step 4: According to the puzzle, 100000 + N is one third of 10 * N + 1, so write the equation: 100000 + N = (10 * N + 1) / 3.Step 5: Multiply both sides by 3 to clear the denominator: 3 * (100000 + N) = 10 * N + 1.Step 6: Expand and simplify: 300000 + 3N = 10N + 1.Step 7: Rearrange terms: 300000 - 1 = 10N - 3N, giving 299999 = 7N.Step 8: Solve for N: N = 299999 / 7 = 42857. Verification / Alternative check: To verify, compute both constructed numbers. With N = 42857, the number with 1 in front is 142857. The number with 1 at the end is 428571. Now check the ratio: 428571 / 142857 = 3 exactly. Thus 142857 is one third of 428571, which confirms that N = 42857 satisfies the condition perfectly. None of the other options gives this exact relationship when checked in the same way. Why Other Options Are Wrong: If you repeat the process with 51428, 57142 or 42168, the resulting numbers 151428 and 514281 (and similarly for the others) will not be in a simple 1:3 ratio. The divisions will not be exact integers, which means those candidates do not meet the condition. Only 42857 produces a pair of six-digit numbers with a perfect factor-of-three relationship. Common Pitfalls: A frequent mistake is to misinterpret the phrase "three times smaller" or to set up the equation incorrectly, for example by making 10 * N + 1 equal to one third of 100000 + N instead of the other way round. Another common error is to attempt trial and error without using algebra, which is tedious and prone to mistakes. Translating the word statement carefully into a clear equation is the most reliable method. Final Answer: The original five-digit number that satisfies the condition is 42857.

Question 16

In the number sequence 0, 10, 1110, 3110, 132110, each term is generated by describing the digits of the previous term. What is the next term that should replace the question mark in this sequence?

Accepted Answer

Introduction / Context: This question is another variation of the look-and-say style sequence, but here the pattern starts from 0: 0, 10, 1110, 3110, 132110, and so on. The idea is that each term is obtained by reading off the previous term as counts of identical digits and then writing those counts and digits as a new number. The problem checks whether you can correctly identify the groups of digits in 132110 and then encode that description into the next term. Given Data / Assumptions: The sequence given is: 0, 10, 1110, 3110, 132110. Each term is formed by "describing" the previous term in terms of runs of identical digits. We must find the term that follows 132110. The answer must be chosen from the provided options, which are all long digit strings. We assume there is no hidden arithmetic beyond counting groups of digits. Concept / Approach: For each term, we group consecutive identical digits in the previous term and then write down the count followed by the digit. For example, starting from 0, there is "one 0", which becomes 10. From 10, there is "one 1, one 0", which becomes 1110. From 1110, we get "three 1s, one 0", which becomes 3110, and so on. We apply this same rule to 132110 to generate the next term. Step-by-Step Solution: Step 1: Write the last known term clearly: 132110.Step 2: Group consecutive digits: the digits appear as 1, 3, 2, 1, 1, 0. The runs are: "one 1", "one 3", "one 2", "two 1s", "one 0".Step 3: Convert each run into a count followed by the digit: "one 1" becomes 11, "one 3" becomes 13, "one 2" becomes 12, "two 1s" becomes 21, and "one 0" becomes 10.Step 4: Write these parts in order: 11, 13, 12, 21, 10.Step 5: Concatenate them into a single number: 11 13 12 21 10 = 1113122110. Verification / Alternative check: To confirm that the same rule has been consistently applied, you can recheck earlier terms. From 3110, the groups are "one 3", "two 1s", "one 0", which gives 132110. That matches the given sequence. Applying the same grouping-and-describing method to 132110 naturally yields 1113122110, confirming that the pattern has been followed correctly. Why Other Options Are Wrong: Option 311322110 misorders the counts and digits, mixing the structure. Option 33122110 suggests there are three 3s or misreads the groups of 1s and 2s, which is incorrect. Option 13222110 incorrectly counts the number of each digit. Only option 1113122110 correctly represents "one 1, one 3, one 2, two 1s, one 0" from the term 132110. Common Pitfalls: Students often misgroup digits, especially when there are multiple single digits in a row, or they try to treat the sequence as an arithmetic or geometric progression rather than a descriptive sequence. Another common error is to miscount the number of repeated digits in the middle of the term. Carefully scanning from left to right, grouping identical digits, and then writing the count followed by the digit helps avoid these mistakes. Final Answer: The correct next term in the sequence is 1113122110.

Question 17

In this classic brain teaser, how many times can you subtract 10 from 100 before the operation no longer matches the original statement?

Accepted Answer

Introduction / Context: This puzzle looks like a simple arithmetic question, but it is actually a trick riddle that tests your careful reading and logical reasoning. At first glance, many people try to compute how many times 10 can be taken away from 100, treating it as a division problem. However, the wording is specific: it asks how many times you can subtract 10 from 100, not from the changing result. Understanding this difference between the first subtraction from 100 and later subtractions from smaller numbers is the key to solving the riddle correctly. Given Data / Assumptions: You start with the number 100. You repeatedly subtract 10 from the current value. The exact wording is "subtract 10 from 100", not "subtract 10 repeatedly until you reach zero". Basic arithmetic rules for subtraction are assumed to be known. Concept / Approach: The phrase "subtract 10 from 100" describes a very specific operation: 100 - 10. You can perform that exact operation only once, because after the first subtraction the number is no longer 100. Later subtractions are taken from 90, 80, 70 and so on, which means you are no longer subtracting 10 from 100 but from other values. The riddle plays on the difference between a strict reading of the expression and the intuitive idea of repeated subtraction or division by 10. Step-by-Step Solution: Step 1: Start with the initial number: 100. Step 2: Perform the first subtraction exactly as stated in the riddle: 100 - 10 = 90. Step 3: Observe that after this step, the current number is 90, not 100. Step 4: If you subtract 10 again, you now calculate 90 - 10, which is 80, so you are subtracting 10 from 90, not from 100. Step 5: Conclude that the operation "subtract 10 from 100" can only be done once, because after that the starting number is no longer 100. Verification / Alternative check: Another way to verify the answer is to compare two interpretations. If the question were purely about how many equal steps of 10 are needed to reduce 100 to 0, you would solve 100 / 10 = 10 and say ten times. But the riddle does not ask "How many times can you subtract 10 in total?" Instead, it literally asks how many times you can subtract 10 from 100. After the first subtraction, the phrase is no longer accurate because the number has changed. This confirms that the logically correct answer is "1 time". Why Other Options Are Wrong: 9 times: This often comes from thinking about how many times you can move from 100 down to 10, but it still ignores the exact wording about starting from 100 each time. 10 times: This is the result of treating the problem like 100 / 10, but later subtractions are from 90, 80, 70, etc., not from 100. 0 times: You definitely can subtract 10 from 100 once, so 0 is incorrect. As many times as you like: While you can keep subtracting 10 from the changing result, you cannot keep subtracting 10 from 100 specifically. The riddle is not about infinite repetition. Common Pitfalls: The most common mistake is to switch unconsciously from a precise language interpretation to a pure arithmetic sequence. Many test takers immediately think "100 divided by 10 equals 10" and choose 10 times without considering that only the first subtraction is truly "from 100". Others may rush and simply guess between 9 and 10 without rereading the puzzle. To avoid this, always pay close attention to wording in verbal reasoning questions and check whether the phrase must remain literally true after each step. Once you get used to this style of riddle, you will spot the trick quickly. Final Answer: You can subtract 10 from 100 only 1 time, because after that, the number is no longer 100.

Question 18

Maria has some blueberries and some cups. If she puts 9 blueberries into each cup, she will have two cups left empty. If she puts 6 blueberries into each cup, she will have three blueberries left over. How many cups and how many blueberries does Mar…

Accepted Answer

Introduction / Context: This puzzle is a nice example of how everyday situations can be translated into simple algebra. Maria has an unknown number of cups and an unknown number of blueberries. Two different filling rules are described, and in each scenario something is left over: either extra cups or extra blueberries. The question asks you to determine both quantities. This type of problem trains your ability to form equations from word statements, an essential skill for quantitative aptitude tests and school mathematics. Given Data / Assumptions: Maria has a certain number of cups; call this number c. She has a certain number of blueberries; call this number b. If she puts 9 blueberries into each cup, exactly two cups remain empty. If she puts 6 blueberries into each cup, she has 3 blueberries left over after filling all cups. We assume all blueberries and cups are identical and that she fills cups fully following the rule in each scenario. Concept / Approach: To solve the puzzle, we convert each description into an equation. When Maria puts 9 blueberries into each cup and leaves two cups empty, she fills (c - 2) cups, each with 9 blueberries. That means b = 9 * (c - 2). In the second scenario, she puts 6 blueberries into each of the c cups, using 6 * c blueberries, and still has 3 blueberries left over, so b = 6 * c + 3. We now have two expressions for b in terms of c, which lets us create and solve a simple linear equation for c, and then compute b. Step-by-Step Solution: Step 1: Let c be the number of cups and b be the number of blueberries. Step 2: From the first scenario, write b = 9 * (c - 2), because only c - 2 cups are filled with 9 blueberries each. Step 3: From the second scenario, write b = 6 * c + 3, because every cup gets 6 blueberries and 3 remain unused. Step 4: Set the two expressions for b equal to each other: 9 * (c - 2) = 6 * c + 3. Step 5: Expand and simplify: 9c - 18 = 6c + 3, so 9c - 6c = 3 + 18, giving 3c = 21. Step 6: Solve for c: c = 21 / 3 = 7 cups. Step 7: Substitute c = 7 into b = 6 * c + 3: b = 6 * 7 + 3 = 42 + 3 = 45 blueberries. Step 8: Check with the first formula: b = 9 * (c - 2) = 9 * (7 - 2) = 9 * 5 = 45, which matches, confirming the solution. Verification / Alternative check: We can verify the answer using a direct reasoning check instead of algebra. Assume Maria has 7 cups and 45 blueberries. If she tries to put 9 blueberries in each cup, she needs 9 * 7 = 63 blueberries to fill all 7 cups. She only has 45, so she can fill 5 cups fully, using 45 blueberries, and 2 cups stay empty, matching the first condition. If she puts 6 blueberries in each of 7 cups, she uses 6 * 7 = 42 blueberries and has 45 - 42 = 3 blueberries left, which matches the second condition. Since both conditions are satisfied, the solution is consistent and unique. Why Other Options Are Wrong: 5 cups and 45 blueberries: With 5 cups, leaving 2 cups empty would not make sense, and the second scenario calculation fails. 7 cups and 39 blueberries: 39 cannot satisfy both the 9-per-cup and 6-per-cup descriptions simultaneously. 9 cups and 45 blueberries: The counts do not fit both conditions; too many cups compared to berries. 9 cups and 57 blueberries: This also fails when you test each scenario step by step; it does not satisfy both conditions at the same time. Common Pitfalls: A common error is to misinterpret "two cups left" as "two blueberries left" or to forget that two different scenarios must be true for the same values of cups and blueberries. Another frequent mistake is to create incorrect equations, such as b = 9 * c + 2, which ignores the wording about empty cups. Carefully translating each sentence into a mathematical expression is crucial. Students should also remember to verify proposed answers with both scenarios instead of trusting algebraic manipulation alone, as a miswritten equation can lead to consistent but wrong numbers. Final Answer: Maria has 7 cups and 45 blueberries in total.

Question 19

In this number pattern puzzle, if 9 = 63, 8 = 48 and 7 = 35, then according to the same rule what should 4 be equal to?

Accepted Answer

Introduction / Context: This question presents an apparent sequence of equalities such as 9 = 63, 8 = 48 and 7 = 35, and then asks what number 4 should correspond to under the same hidden rule. Problems like this are classic number pattern puzzles that test your ability to detect relationships rather than perform complicated calculations. The key is to look for a simple arithmetic operation that consistently turns the left hand number into the right hand number for all given examples. Given Data / Assumptions: The pattern shows: 9 corresponds to 63, 8 corresponds to 48, and 7 corresponds to 35. The question asks for the value that 4 should correspond to under the same rule. We assume standard arithmetic operations like addition, subtraction, multiplication and division. The same rule must work for all given pairs, not just one of them. Concept / Approach: A good strategy for these puzzles is to check simple expressions such as n * (n - 1), n * (n - 2) or 2 * n^2 and see if any match all entries. For 9 = 63, the product 9 * 7 equals 63, and 7 is 9 - 2. For 8 = 48, the product 8 * 6 equals 48, and 6 is 8 - 2. For 7 = 35, the product 7 * 5 equals 35, and 5 is 7 - 2. This reveals the pattern: each number n is being multiplied by (n - 2). Once we identify this rule, we can apply it to 4 to find the missing value. Step-by-Step Solution: Step 1: Consider the first pair: 9 and 63. Compute 9 * (9 - 2) = 9 * 7 = 63. Step 2: Check the second pair using the same rule: 8 * (8 - 2) = 8 * 6 = 48, which matches the given value. Step 3: Check the third pair: 7 * (7 - 2) = 7 * 5 = 35, again matching the given value. Step 4: Conclude that the rule is "multiply the number by two less than itself" or n * (n - 2). Step 5: Apply the rule to 4: compute 4 * (4 - 2) = 4 * 2 = 8. Step 6: Compare 8 with the answer options and select the matching choice. Verification / Alternative check: To verify, we can test whether any alternative rule might produce the same sequence without contradicting the data. For example, n^2 - 2n also gives n * (n - 2), so it is essentially the same method and not a new rule. No simple linear rule like 7n or 5n + something fits all three given pairs at once. Because the pattern n * (n - 2) correctly reproduces every example and gives a clear value for 4, we can be confident that it is the intended logic of the puzzle. Therefore, the answer 8 is unique and consistent. Why Other Options Are Wrong: 4: This would require a rule that shrinks the number instead of multiplying it by a positive factor, which does not match the earlier pairs. 12: 4 * 3 equals 12, but that would imply multiplying n by n - 1, which fails for 9 and 8 because 9 * 8 is not 63 and 8 * 7 is not 48. 16: 4 * 4 equals 16, which suggests squaring the number, but 9^2 is 81, not 63, and 8^2 is 64, not 48. None of these: This is incorrect because 8 fits perfectly with the established pattern n * (n - 2). Common Pitfalls: Many learners try complicated formulas too early, such as combining squares, cubes or multiple operations, when a simple multiplication pattern is enough. Another common mistake is to find a rule that fits only one or two of the given examples and then stop checking. A valid pattern must apply to all provided pairs. Some students also guess based on the size of the numbers instead of verifying calculations. To avoid these issues, always test your rule against every example in the puzzle and confirm that it works consistently before applying it to the missing case. Final Answer: Using the rule n * (n - 2), the value corresponding to 4 is 8.

Question 20

In a certain city, 5% of all persons have unlisted phone numbers. If you randomly select 100 names from that city's phone directory, how many of the selected people will have unlisted phone numbers?

Accepted Answer

Introduction / Context: This puzzle looks like a straightforward percentage question but is actually testing careful reading and logical understanding of what a phone directory contains. You are told that 5% of all people in the town have unlisted phone numbers, and then you are asked about a sample taken from the city's phone directory. The trick is to notice that a phone directory, by definition, lists only people with listed numbers. People with unlisted numbers do not appear in that directory at all, so the usual percentage calculation does not apply in the way many students first expect. Given Data / Assumptions: In the town as a whole, 5% of persons have unlisted phone numbers. The remaining 95% have listed phone numbers and therefore appear in the directory. You select 100 names at random from the city's phone directory, not from the entire population of the town. The directory contains only those people whose numbers are listed. Concept / Approach: The core idea is to understand the difference between selecting from the entire population and selecting from a filtered list. The 5% figure is a property of the whole population. However, the phone directory is already a filtered subset that excludes everyone with an unlisted number. When you randomly choose names from this directory, you are sampling only from the listed group. As a result, the probability of picking someone with an unlisted number from the directory is actually zero, because such people are not present in that list. Step-by-Step Solution: Step 1: Interpret the statement "5% of all persons have unlisted numbers" as a property of the entire town, not of the directory itself. Step 2: Recall that a phone directory is a published list of people whose telephone numbers are listed and available to the public. Step 3: Recognise that people with unlisted numbers request that their details not be printed, so they are completely absent from the directory. Step 4: Notice that the question specifies selecting 100 names from the city's phone directory, meaning every chosen name must belong to someone with a listed number. Step 5: Conclude that none of the 100 selected people will have unlisted phone numbers, because unlisted numbers never appear in the directory in the first place. Verification / Alternative check: To double check, imagine a small town with 1000 people. If 5% have unlisted numbers, then 50 people are unlisted and 950 are listed. The directory would contain 950 names and omit the 50 unlisted people. If you now pick 100 names at random from this directory of 950 entries, each chosen entry must be one of the 950 listed people, not one of the missing 50. Therefore, the count of unlisted numbers in your sample is exactly 0. This reasoning does not depend on the specific numbers; it simply uses the fact that unlisted people are not printed in the directory at all. Why Other Options Are Wrong: 5: This assumes you are taking 5% of 100 directly, but that would only be valid if you were sampling from the entire town, not from the directory. 95: This misinterprets the 95% as being unlisted instead of listed, reversing the given information. Cannot be determined: The information is actually sufficient. Once you understand that unlisted numbers never appear in the directory, the answer is determined exactly. 10: This is another guess based on percentages, but there is no basis for doubling or changing the 5% figure in the problem statement. Common Pitfalls: The main pitfall is to see percentages and immediately multiply 5% by 100 without considering what the 100 represents. Many students overlook the crucial detail that the sample is taken from the phone directory, not from the town population. Others may think that unlisted numbers somehow appear in the directory but are hidden, which is not how directories work in standard reasoning problems. To avoid such errors, always ask, "From which group am I sampling?" and verify whether that group already excludes or includes certain categories of people. Careful attention to the sampling frame is an important skill in both statistics and logical reasoning. Final Answer: Because the directory includes only listed numbers, 0 of the 100 selected people will have unlisted phone numbers.

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