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?

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.