How do you solve cryptarithms?

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tiburtius's profile pic

tiburtius | High School Teacher | (Level 2) Educator

Posted on

Let me give you an example. Probably the most famous cryptarithm:

send+more=money

In cryptarithms like this it's good to search for digits 0 and 9 first. Since money is one digit longer than send and more it follows that first digit m must be 1 that is "carry 1". So now we have:

send+1ore=1oney

Here we have s+1=10 or s+1=11 in the later case we would have o=1 which cannot be true since we have already establish that m=1, thus o=0. Now we have:

send+10re=10ney

Here we have s+1=10 so s can only be equal to 9  or 8. But if s=8 that would mean that we have "carry 1" from addition of previous digits namely e+0=10 (we can have at most 10 only if e=9 and we have "carry 1" from previous addition) which cannot be because we would have n=0 but we have already establish that o=0. Thus we have s=9.

9end+10re=10ney

Now we have e+0=n meaning that n is by one greater than e i.e. n=e+1   (1)

Also because we must have "carry 1" from previous digits we have n+r+(1)=e+10    (2)

(1) is in brackets because we still don't know if we have "carry 1" from previous digits). Now we subtract (2)-(1) and get

r+(1)=9

And since we've establish that s=9, r cannot be 9 so it must be r=8. And we now know that we have "carry 1" from previous (first) digits.

9end+108e=10ney

Now for the fist digits d+e>12 (because we have "carry 1" so it must be >10 and y cannot be 0 or 1 because m=1 and o=0). So what digits can we have for d and e? We can't have 9 or 8 (s=9 and r=8). We can only have 7 and 6 or 7 and 5. Now we remember equation (1) and see that n lies between d and e. Hence we have e=5, n=6 and d=7. And also d+e=10+y gives us y=2. So the solution is

9567+1085=10652

which is true.

There is no general way for solving such problems but you usually start by searching for 0, 1 or 9 and after that you try some logic and basic arithmetic and if you must trial and error. 

baxthum8's profile pic

baxthum8 | High School Teacher | (Level 3) Associate Educator

Posted on

How do you solve cryptarithms?


Cryptarithms are a type of mathematical puzzle in which the digits are replaced by symbols (typically letters of the alphabet). For example:

9567 + 1085 = 10652

can be represented like this:

abcd + efgb = efcbh

Cryptarithm Rules

  • Each letter represents a unique digit.
  • Numbers must not start with a zero.
  • The solution is unique (unless otherwise stated).
Sources:

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