The sixth grade math puzzle - First example: Number at the top: 5, number at the left: 3, number at the right: 2, number at the bottom: 4, number at the center: 19. Second example: Number at the top: 9, number at the left: 3, number at the right: 3, number at the bottom: 7, number at the center: 54. Puzzle: Number at the top: 6, number at the left: 5, number at the right: 4, number at the bottom: 8, number at the center: ?

I recently came across this puzzle on Reddit, where the parent of a sixth grader confessed that after spending many hours, they could not crack the puzzle. I must admit that it took me a few minutes to come up with the solution.

Now, not many of my readers are in the sixth grade. So, I made one small modification to the puzzle. The original puzzle featured 4 choices for the potential answer. In the version that I am presenting today, I have removed the multiple choices.

If it is any consolation, when I came up with my solution, I did not look at any of the options (they were of not much help to me).

Spoiler Alert

Beyond this section, I will be explicitly discussing the solution to this puzzle. If you wish to solve this puzzle on your own, I recommend you pause at this point and give it a try.

Once you are done with your attempt, you may continue reading to compare our approaches. All the best.

Also, if you find yourself struggling, you may check out the optional hint below.

Optional Hint

When I started solving the puzzle, I initially over-complicated things by considering complex mathematical tools. But then, I reminded myself that this puzzle was originally designed for a sixth grader.

What could be the mathematical tools available to a sixth grader to solve this puzzle?

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Setting Up the Puzzle

By just looking at the puzzle, we can say that the four numbers in the periphery are the inputs and the number at the center is the output. The first two iterations present examples from which we could establish a pattern.

The sixth grade math puzzle — First example: Number at the top: 5, number at the left: 3, number at the right: 2, number at the bottom: 4, number at the center: 19. Second example: Number at the top: 9, number at the left: 3, number at the right: 3, number at the bottom: 7, number at the center: 54. Puzzle: Number at the top: 6, number at the left: 5, number at the right: 4, number at the bottom: 8, number at the center: ?
The sixth grade math puzzle — Illustration created by the author

The third iteration requires us to compute the output number (marked by the question mark), given the four input numbers.

To make our lives easier, let me start by assigning four variables to the input numbers as follows:

1. The number at the top — T

2. The number at the left — L

3. The number at the bottom — B

4. The number at the right — R

The sixth grade math puzzle setup — Example: Number at the top: T, number at the left: L, number at the right: R, number at the bottom: B, number at the center: ?
Puzzle setup — Illustration created by the author

Given this setup, we are all set to solve this puzzle.

Solution to the Puzzle

Considering that this puzzle was designed for a sixth grader, we could simply start with patterns that use the four fundamental arithmetic operations — addition, subtraction, multiplication, and division.

In my first attempt, I came up with the following pattern:

(T*B) – (L-R)

If we apply this to the first example, we get (5 * 4) – (3 – 2) = 20 – 1= 19. That is a good start. But then, when we apply this pattern to the second example, we get (9 * 7) – (3 – 3) = 63 – 0 = 63.

In my first attempt, I came up with the following pattern: (T*B) — (L-R) If we apply this to the first example, we get (5 * 4) — (3–2) = 20–1= 19. That is a good start. But then, when we apply this pattern to the second example, we get (9 * 7) — (3–3) = 63–0 = 63.
First attempt to solve the puzzle — Math illustrated by the author

This is obviously not the correct pattern. But just based on the proximity of 63 to 54, I would guess that I am close. At least, starting with the basic arithmetic operations was the right approach.

In my second attempt, I came up with the following pattern:

(L*B*R) – T

When we apply this pattern to the first example, we get (3 * 4 * 2) – 5 = 24 – 5 = 19. Next, when we apply it to the second example, we get (3 * 7 * 3) – 9 = 63 – 9 = 54. Bingo! Now we have a pattern that matches both the given examples.

In my second attempt, I came up with the following pattern: (L*B*R) — R When we apply this pattern to the first example, we get (3 * 4 * 2) — 5 = 24–5 = 19. Next, when we apply it to the second example, we get (3 * 7 * 3) — 9 = 63–9 = 54. Bingo! Now we have a pattern that matches both the given examples.
Second attempt to solve the puzzle — Math illustrated by the author

Finally, when we apply this pattern to inputs of the unknown output, we get (5 * 8 * 4) – 6 = 160 – 6 = 154. There you go! The correct answer is 154.

Finally, when we apply this pattern to inputs of the unknown output, we get (5 * 8 * 4) — 6 = 160–6 = 154. There you go! The correct answer is 154.
Solution to the puzzle — Math illustrated by the author

I hope you had fun solving this puzzle. It has been a while since I wrote about puzzles. A few readers like Avi Kotzer said that they missed my columns/essays on puzzles. So, I will try to write more about intriguing puzzles in the future.

If you are into that sort of stuff, be sure to check back here in the future!

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Further reading that might interest you: 

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