Can You Really Solve This Third-Grade Math Puzzle?

I recently came across a third-grade math puzzle that made me think. My first thought was that I shouldn’t be thinking this much about it. My thought continued:

“Either I am so bad in math or this puzzle is too complex for a third-grader.”

To make things fun for myself, I decided to assume that the puzzle applies to people beyond the third grade, and continued solving it. In the end, I managed to arrive at a reasonable solution. In this essay, I will be going through the process that I followed to arrive at thesolution.

As you read along, I will explicitly mention a ‘spoiler alert’ at a point where you may choose to try and solve the puzzle on your own.

This essay is supported by Generatebg

Intuition For Input-Output Puzzles

Whenever we see such input-output puzzles, we (as human beings) tend to look for a pattern. Such a pattern would explain how each input leads to the corresponding output.

We would then try to formulate this pattern as a mathematical rule or formula for the given input variable(s). However, in this case, things seem to be a little tricker. A pattern is not so obvious.

In such cases, what I usually do is visually plot the given inputs and outputs. In this case, the visual plot turns out to be the following:

Once I obtained the visual plot, it became clear to me why I thought that this puzzle was beyond a third-grader. And therein lies the clue for you to solve this puzzle as well.

Before we proceed towards solving this puzzle, just know that there are an infinite number of possible solutions for such a puzzle if we consider discrete maps (rules) for each input-output combination. Instead, the challenge I pose to you is ‘continuity’ between the various input-output combinations. Using the visual plot as your clue, how would you proceed?

Spoiler Alert: From this point on, I will be explicitly discussing the solution. So, you may choose to move away from this essay and try to solve the puzzle on your own. Later, you may come back to see if your approach indeed matches mine.

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Solution to the Third-Grade Puzzle

Remember that we had difficulty intuitively sensing any pattern when we first looked at the input-output table? This is because human beings are typically linear thinkers. Had the pattern been a straight line, we would have picked it up immediately.

After we visually plot the input-output combinations, it becomes apparent that the relationship (or the pattern) is non-linear. That is, the three given points do not pass through a straight line. In other words, we could assume that a polynomial solution exists.

The Polynomial Equation Set

The generic polynomial equation is given by the following expression:

y = ax² + bx + c

where a,b, and c are arbitrary constants.

When we plug in the given x (input) and y (output) values into the above equation, we obtain the following equation set:

So, we have three equations with three unknowns. This can be solved relatively quickly using algebraic substitution.

Solving the Equation Set

When we subtract equation 1 from equation-2, we eliminate ‘c’ and end up with equation-4. Similarly, when we subtract equation-2 from equation-3, we obtain equation-5 that is comparable with equation-4.

Now, we can just subtract equation-4 from equation-5 to obtain the value of ‘a’ as follows:

Now that we have the value of ‘a’, we could plug this back into equation-4 to get the value of ‘b’ as follows:

When we plug the values of a and b into equation-1, we get the value of ‘c’ as follows:

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The Final Solution to the Third-Grade Puzzle

To obtain the final solution to our ‘third-grade’ puzzle, we just plug back the values of ‘a’, ‘b’, and ‘c’ into the original polynomial equation for ‘x = 10’ to obtain y:

There you have it. That is the answer to our puzzle.

Closing Thoughts

It is quite clear that this puzzle is beyond the third-grade level. My guess (and the consensus) is that there must have been a typographical error in the original question.

If we change the output for the input ‘3’ to be ‘15’, we would establish a linear relationship. This would make the puzzle a lot more intuitive to solve for your typical third-grader.

Having said that, I just used the original puzzle as an excuse to do some math and whip out some hand-drawn illustrations. I hope you enjoyed the ride as much as I did!

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Source and Credit: r/HomeworkHelp and Presh Talwalkar.

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Further reading that might interest you: How To Calculate The Day Of The Week For Any Date? and How Easy It Is Really To Predict The Future?