[This is a back-issue of this website’s newsletter]
A while ago, I promised I’d put some cryptarithm puzzles onto Dr Mike’s Math Games for Kids. Well, I will, but they may only get there next year. Things have been busy!
If you don’t know what a cryptarithm puzzle is, it’s a puzzle where all the digits in an arithmetic sum have been replaced by letters – each letter corresponding to a unique digit. They are sometimes called alphametics, crypt-arithmetic puzzles, or even letter arithmetic or verbal arithmetic puzzles. The classic examples is SEND + MORE = MONEY… each digit in a correct sum has been replaced by a letter. Your job is to figure out the original sum!
I wrote a program to find some cryptarithms to put on the site, and it found four million of them! Rather than just pick a few for the website and throw away the rest, I organized all four million onto a CD. The CD is now on sale for US$14.99. Sometime early next year, I’ll pick a selection from the CD and make them available for free on Dr Mike’s Math Games for Kids – or, you can submit your email address here and get a selection for free by email over the next month.
Another thing I promised almost a year ago is that I’d explain the amazing link between this game, magic squares, and tic-tac-toe. In the game, players take turns to select numbers from 1 to 9. Once a number is picked, it cannot be picked again. The winner is the first to have three numbers that add up to 15. For want of a better name, I call it the “1 to 9 15 game”. A lousy name, I know!
To see the amazing link, do this :
- Take a tic-tac-toe board, and write a 3×3 magic square in it – lightly, with pencil.
- Play the 1 to 9 15 game
- Each player, as they take their turn, crosses off the number they chose from the magic square. One player crosses them off with X’s, the other with O’s.
- The winner, by the rules of the game, is the first player to get three numbers summing to 15. As it happens, this will also be the first person to get three in a row on the tic-tac-toe board…
So, the 1 to 9 15 game is exactly the same game as tic-tac-toe, but in a disguised form. They are equivalent games! Any good tic-tac-toe player who knows their magic squares can become an instant expert at the 1 to 9 15 game…
This idea that two apparently different things can actually be the same if you look at them the right way is a very important one in real mathematics. Some examples most math students meets before college include
- The link between differential and integral calculus – that finding areas can be the same as “un-differentiating”
- Combinations and Pascal’s triangle – that simple addition can be used for counting combinations
- The number “pi” and the fact that it appears in all sorts of places in mathematics
- Even the basic idea behind all algebra – that an unknown number can be represented by a letter
And that, in a way, brings this email full circle back to cryptarithms!
Well, that’s all for this email – have a very happy Christmas (or just a happy holiday time, if you don’t celebrate Christmas), and best wishes for the New Year!
Yours,
Michael Hartley