Cryptarithms

I recommend reading this before watching the video…

What are cryptarithms?

Cryptarithms, sometimes known as alphametics, are puzzles where you are given an arithmetical expression where the digits have been replaced by letters, each digit a different letter. Your job is to ‘crack the code’, so to speak, to find out what is the digit that each letters represent.
Just for fun, many of the times, the letters actually spell words. One of the most famous alphametics, spelling out ‘SEND MORE MONEY’ was first published by Henry Dudeney, a British puzzlist, in 1924.

Five rules govern alphametics:

  1. Identical digits are replaced by the same letter. Different digits are replaced by different letters.
  2. After replacing all the letters with digits, the resulting arithmetic expression must be mathematically correct.
  3. Numbers cannot start with 0. For example, the number 0900 is illegal.
  4. Each problem must have exactly one solution, unless stated otherwise.
  5. The problems will be in base 10 unless otherwise specified. This means that the letters replace some or all of the 10 digits – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Solving cryptarithms

There are key elements to solving most alphametics.

  • In many cases the result of an addition problem is one digit longer (in digit-length) than the addends - the numbers added. If there are only two addends, this implies that the extra digit is the number 1.

Let’s look at a very simple alphametic: ME+ME=BEE

The letter B must represent the digit 1, since when you add two 2-digit numbers you cannot possibly get a number larger than 198. That happens when both addends are 99. Since M and E are two different numbers, they will certainly be even smaller than 99! In any case, the hundreds digit in the sum, represented by B in our example, must be 1.

  • In two addend alphametics, there may be columns that have the same letter in both the addends and the result. If such a column is the units column, that letter must be 0. Otherwise, it can either be 0 or 9 (and then there is a carry).

  • If there are more than 2 addends, the same rules apply but need to be adjusted to accommodate other ‘carry’ possibilities. If there are 3 addends and an ‘extra’ digit in the result, this digit can now be, 1 or 2. If there is a column with the same letter, this letter can now be: 0,1 or 2, and so on.

In the alphametic: ME+ME-BEE the column of the unit’s digits is: E+E=E There is only one digit, which has the property that when you add it to itself you get the same digit as the result – zero! Only the sum of two zeros is zero, so E must be equal to 0.

The solution to this alphametic is therefore: B=1, E=0, M=5: 50+50=100.

Now watch the video!

Now let’s look at a slightly more advanced cryptarithm. This video shows how to solve the alphametic: NO + GUN + NO = HUNT. Note the ‘neat’ sentence: “No gun, no hunt!”.

Challenge your fellow learners

Post your own cryptarithms in the comments below, but don’t give away the answers!

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This video is from the free online course:

An Introduction to Recreational Math: Fun, Games and Puzzles

Weizmann Institute of Science