One of the goals of creating this blog was to show you some cool math stuff you might not have seen before. So I thought I’d create a puzzle about this:
Just replace each letter with a digit from 0–9 so that the sum is correct. No number begins with a 0. One more thing: M + T = A. Good luck!
Perhaps you’ve never seen puzzles like this before — they’re called cryptarithms. At first they look impossible to solve — almost any assignment of numbers to letters seems possible. But not really. You’re welcome to try on your own first — but feel free to read on for some helpful advice.
Look at the last column (the units). If L + H + G ends in H, then L + G must end in a 0. Since L and G are digits, then L + G = 10. This doesn’t completely determine L or G, but once you know one of the numbers, you know the other.
As a result, you also know there’s a carry over to the third column (tens). What does this mean? That 1 + O + T + O ends in C. Since O + O is an even number, this means that if T is even, then C is odd, and if T is odd, then C is even. We even know a bit more about T from looking at the sum: T is either 1 or 2, since adding three numbers less than 10,000 gives a sum less than 30,000.
What about the second column? O + A + L ends in A. We might be tempted to think that O + L must end in a 0, but that would mean that O + L is 10. This can’t be, since that would mean that G = O (remember that L + G = 10). Therefore there has to be a carry from the third column over to the second column, meaning O + L = 9. So O is one less than G.
Get the idea? There’s a lot of information you can figure out by looking at the structure of the letters in the sum. But it turns out that without the condition M + T = A, there are 12 solutions to this puzzle! Multiple solutions must occur here, for if you can solve this puzzle, you can also solve
The M and B occur just once, and at the beginning of numbers. This means if M = 7 and B = 9 in a solution, then putting M = 9 and B = 7 — with all other letters staying the same — will also produce a solution. In looking at all the solutions, I found that giving M + T = A results in a unique solution without giving values for specific letters.
That’s all the help you get! Sometimes you might just guess well and stumble onto a solution — but take the additional challenge and prove you’ve got the only solution to the cryptarithm.
How do you create a cryptarithm? There was a time I did so by hand — but those days are gone. Read more if you’d like to see how you can use programming to help you create these neat puzzles. (Of course there are online cryptarithm solvers, but that takes all the fun out of it!)
Creativity in Mathematics 