I saw this posted on Facebook yesterday:

Several people posted that the answer to this question is 90. Others said 72, 36, 0 or some other number. None of them were wrong.

This is a logic problem that does not provide enough information to explain it with only one correct solution. Several answers can satisfy the question as it is presented. How valid those answers are is debatable. For example, here are two solutions that are questionable:

In the first example on the left, the proposed solution is that each number is multiplied by a number that increases by 1 each time. It looks simple, but the numbers on the left skip from 6 to 9. Should this absence be taken into account when writing the multiplier numbers on the right? This solution opts to ignore the skipped numbers, so 9 is multiplied by 8, and the answer is 72.

The solution on the right suggests multiplying each number by the number below it to get what the top number equals. This approach does take into the account that the numbers 7 and 8 are skipped. In fact, this solution does not work if they’re not taken into account, as 6 x 9 equals 54, not 42. Further, this solution assumes that the number below 9 will be 10, giving the result of 90. Some people responding to this problem on Facebook argued that the numbers could start over again at 0 or 1, or perhaps wrap around from 9 back to 2. These assumptions would give the answers 0, 9, and 18 respectively. Any of these solutions could be considered correct.

Returning to the solution on the left, let’s look at how it changes if we take the skipped numbers into account. Then we can write the solution in this way:

In this approach, the answer becomes 90, not 72.

This solution can also be written as a mathematical formula:

Using this formula, we find the answer of 90 again:

What I like about using this formula as the solution is that it no longer matters if we add 7 and 8 into the solution or not; the formula still gives the result of 90. However, this is not the only solution that works whether we consider the missing 7 and 8 or not. We could instead use the equation X*||X-6|-7|, which provides an answer of 36:

These are not the only solutions to the proposed problem. They are a few examples of how this problem can be interpreted and solved.

The respondents on Facebook could not agree on what the correct answer would be, and many argued that their proposed answer was the only correct one. Many resorted to name-calling and belittling to make their points, so I won’t be providing a link to that discussion.

Thinking about this problem led me to wonder, not for the first time, what is reality? Even from the rigorous perspectives of math and logic, sometimes the same problem yields different answers. Growing up, I liked the idea that science could boil our reality down to hard truths, absolutes that we can rely upon when making decisions. This isn’t always true, though. Some things cannot be quantified. Some problems have multiply answers that are equally valid. How do we choose between them?

We’re not accustomed to thinking both sides can be right on a given issue. What if they can be? What if some of our most complex problems in life are a result of multiple solutions being equally valid? Our politics are polarized today, but what if the various parties are all presenting effective solutions to the problems we face? How do we choose among them? We argue about religion, spirituality, and the physical world as if only one idea can be true. What if everyone is correct? Is it possible that the truth isn’t black-and-white, or grey? It could be all three simultaneously.

Perhaps I’m being too radical in my conclusions, but it is interesting to contemplate. Could we achieve more harmony and happiness among groups with different values and priorities if we understand that each group could be “right” in their point of view? The truth may be less absolute than we like to think of it as being, and sometimes reality may genuinely be a matter of perspective.

What are your thoughts on this topic?