How to solve alphametic puzzles

I recommend reading this before watching the video… 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).

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.

Here are some tips for solving more complicated alphametics.

  • 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.

  • It is wise to turn subtraction problems into addition problems by adding the result to the smaller addend to get the larger one.
  • When faced with a few options for a letter, try one out until you either get the correct answer, or find a contradiction.

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!”.

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This video is from the free online course:

Maths Puzzles: Cryptarithms, Symbologies and Secret Codes

Weizmann Institute of Science