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 possibilities. If there are four identical letters in the units column (one of them the sum), this letter can now be: 0 or 5 (because 5+5+5=15). If there are four identical letters in a different column (one of them the sum), this letter can now be: 0 or 5 (no carry), 4 or 9 (carry 2). Four identical letters in a column other than the units column means a 1 could not have been carried over (why not?). This rule can be worked out for more than 3 addends as well…
- 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!”.
© Davidson Institute of Science Education, the educational arm of the Weizmann Institute of Science