Do Nothing
Puzzle. How can you make the following equation correct without changing it: 8 + 8 = 91?
The intended answer: turn the paper over! When flipped upside down, the equation becomes 16 = 8 + 8.
As you might expect, my blog post doesn’t stop there. I’d like to share some creative ideas my students came up with when they tackled this puzzle as part of their homework.
The most common suggestion was to interpret the equation modulo some number. For example, it works modulo 75. By extension, it also works modulo any divisor of 75: 3, 5, 15, or 25.
They also suggested interpreting the equation in base 5/3.
One far-fetched but imaginative submission proposed the following: Suppose the equation is written in an alien language whose symbols look identical to ours but have different meanings. In this alien base-10 system, the symbols + and = mean the same as on Earth, but an 8 represents 6, a 9 represents 1, and a 1 represents 2. Then the alien equation 8 + 8 = 91 translates to 6 + 6 = 12 in human, which is perfectly true.
But my favorite answer was the following:
- Interpret the question mark as a variable and solve the equation. This gives ? = 16/91. We didn’t change the equation — just solved it!
Share:
Ivan:
When flipping the paper “upside down” let’s not forget about “This side up” labels on the boxes. In all other cases (flipping around axes X or Y), even if the paper is transparent, we will not produce meaningful results.
12 October 2025, 11:53 pmBernardo Recamán:
The sum of the ages of my three boys, all different ages, is equal to the product of my three daughters’ ages; vice versa, the sum of my three daughters’ ages is equal to the product of my three sons’ ages.
How old is Nur, the eldest of my children?
13 October 2025, 10:38 amIvan:
Since one of the twins is always the eldest, Nur could be 3 years old.
14 October 2025, 3:37 am1*2*3 = 1+2+3
Mesut:
I liked all the answers except the intended answer. However, the notch on the number 1 poses a problem for the intended answer.
31 October 2025, 4:20 pm