Cryptarithms – Creativity in Mathematics

In going through some folders in my office the other day, I came across some sets of mathematics puzzles I wrote for a conference of the International Group for Mathematical Creativity and Giftedness in 2014. Teachers of mathematics of all levels attended, from elementary school to university. The organizing committee (which included me) thought it might be fun to have some mathematical activity that conference attendees could participate in.

So I and my colleagues created three levels of contests — Beginning, Intermediate, and Advanced — since it seemed that it would be difficult to create a single contest that everyone could enjoy. But I did include three problems that were the same at every level, so all participants could talk about some aspect of the contests with each other.

Participants had a few days to get as many answers as they could, and we even had books for prizes! Many remarked how much they enjoyed working out these puzzles.

Now this conference took place before I started writing my blog. I have written several similar contests over the years for various audiences, and so I thought it would be nice to share some of my favorite puzzles from the contests with you. And so The Puzzle Archives are born!

First, I’ll share the three puzzles common to all three contests. I needed to create some puzzles which were fun, and didn’t require any specialized mathematical knowledge. As I’m a fan of cryptarithms and the conference took place in Denver, I created the following puzzle. Here, no letter stands for the digit “0.”

Day133Puzzle1

For the next puzzle, all you need to do is complete the magic square using the even numbers from 2 to 32. Each row, column, and diagonal should add up to the same number. There are two solutions to this puzzle — and so you need to find them both!

Day133Puzzle2

And of course, I had to include one of my favorite types of puzzles, a CrossNumber puzzle. Remember, no entry in a CrossNumber puzzle can begin with “0.”

Day133Puzzle3

I also included a few geometry problems, staples of any math contest. For the first one, you need to find the area of the smallest circle you could fit the following figure into. Both triangles are equilateral; the smaller has side length 1 and the larger has side length 2.

Day133Puzzle4

And for the second one, you need to find the radius of the larger circle. You are given that the smaller circle has a diameter of 2 units, and the sides of the square are 2 units long. Moreover, the smaller circle is tangent to the square at the midpoint of its top edge, and is also tangent to the larger circle.

Day133Puzzle6

The last two problems I’ll share from this contest are number puzzles. The first is a word problem, which I’ll include verbatim from the contest itself.

Tom and Jerry each have a bag of marbles. Tom says, “Hey, Jerry. I have four different colors of marbles in my bag. And the number of each is a different perfect square!” Jerry says, “Wow, Tom! I have four different colors of marbles, too, but the number of each of mine is a different perfect cube!”

If Tom and Jerry have the same total number of marbles, what is the least number of marbles they can have?

And finally, another cryptarithm, but with a twist. In the following multiplication problem, F, I, N, and D represent different digits, and the x‘s can represent any digit. Your job is to find the number F I N D. (And yes, you have enough information to solve the puzzle!)

Day133Puzzle5

Happy solving! You can read more to see the solutions; I didn’t want to just put them at the bottom in case you accidentally saw any answers. I hope you enjoy this new thread!

(Note: The FIND puzzle was from a collection of problems shared by a colleague. The first geometry problem may have come from elsewhere, but after four years, I can’t quite remember….)
Continue reading The Puzzle Archives, I

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!)

Continue reading Cryptarithms