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.
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