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Fill In the Number Puzzles

September 2, 2008

Danielle posted a question on my About page that I thought I’d answer in a separate post.  She wrote:

Hi MathMom!
I have a question… my daughter gets number triangles for math warm-up. E.g. Use each number only one time (1-9) and the sum of the numbers along each side of the triangle must equal 20.

She’s been doing them by trial and error, but that takes quite a long time the more complicated they get. Is there a formula or method that she’s missing?

Thank you for any help you can give her!



There isn’t really a formula that can be used to solve generic puzzles like this, but the key to solving them is thinking about which numbers get used more than once.  In a triangle puzzle, this is the numbers in the corners.  Each of these numbers gets included in 2 of the sums, while the rest of the numbers get included only once.

In the example you gave above, your daughter should first think about what the sum of all the available numbers is if they were each used only once: 1+2+…+8+9 = 45.  Since we need the total of the 3 sides to be 60 (3×20) we know that the numbers that are used twice must add up to 15 (the difference between 60 and the 45 we’d have if each number were used just once).  So, the corners of the triangle should add up to 15.  Unfortunately, there are a lot of ways of making 15 out of 3 of the numbers from 1 to 9.  But it does cut down the trial and error a fair bit to start.  I tried a few combinations and was able to find answers pretty quickly using that constraint.  I hope this insight makes these puzzles a little more fun and less frustrating for your daughter, Danielle!

Here’s another type of fill-in puzzle.  You don’t use sums to solve it, but the key to it rests in realizing which spots are “special”.  I’ll leave it at that for now so as not to give it away for those who’d like to try to solve it.

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