How to solve Cryptarithmetic Puzzles?

by Carson
What are Cryptarithmetic Puzzles?

Sometimes, we see tricky math puzzles that involve turning the numbers into letters or symbols, and they usually have a meaning. So, what is cryptarithmetic puzzle (aka verbal arithmetics or cryptarithms), and how to solve them?


Cryptarithmetic puzzles are problems that have arithmetic operations for things other than numbers. For example, ABCD * 4 = DCBA. Do you have any ideas? If you don’t, don’t be concerned. Solving these problems requires lots of experience and analyzing skills. The simple method will be explained below.

Normally, each letter represents a different number. For instance, 2178 * 4 = 8712, which doesn’t overlap and precisely matches the question in the format. Also, the phrases in the questions usually have meanings, like CRACK + HACK = ERROR (42641 + 9641 = 52282).

How to solve Cryptarithmetic Puzzles?

Like its namesake, it is a simple encryption method because we can’t solve these problems instantly unless we remember the answers. Now, here’s a guide to solve those complicated problems. Probably all you need is a critical concept: List all numbers possible for the figure and eliminate them one by one. However, when you are stuck and have only a handful of choices, you’d probably need to make educated guesses.

First of all, don’t place 0 in the first digit of a number because it decreases the amount of digits on a number, consequently making the problem unsolvable. Moreover, look for identical letters when making any modifications. Furthermore, check all suggestions before you proceed because there may be only one correct answer. Or, when you encounter sums with a digit more than both numbers that make up it, the extra digit can only be 1.


Then, please use your simulation techniques. For example, in the crack-hack-error cryptarithm, the thousands of the first number in the equation and the sum is “R”. Therefore, you need to find integers for “H” that, when added, can have the same number in the sum’s corresponding digit. And these are 0 and 9.

However, consider that “H” is in the header of the integer, and as a result, it can’t be zero. Sounds like playing a Sudoku Game, right? Cryptarithmetic puzzles are just like that.

Guess the Numbers!

However, if you can’t proceed because of a failed step, or multiple options are available without anything that can eliminate them, guess! It’s not like a Sudoku game that every action can have reasons since cryptarithmetic puzzles sometimes don’t have any hints. So, expect to fail every time you execute this procedure.

But, these guesses can take ages if you don’t use the right algorithm. For instance, in the previous cryptarithm, we need an “ACK + ACK = 1ROR” pattern, and we need educated guesses. Once even a single maneuver go wrong, we need to retake the entire process to conserve time. Of course, retain a record in case the error isn’t natural.

For instance, if you predict any solution (like 641 + 641 = (1)282) is valid, test in the Scientific Method. This process is hypothesizing. After that, experiment by initiating a chain reaction: Decode the information or make more educated guesses on other unknown digits. Remember, if you have “sub-guesses”, make sure to disprove all of them before formally rejecting the hypothesis.

If you reach the correct answer (641 + 641 = 1282), your chain reaction will go successfully without any obstacles. Also, care about other parameters before concluding that this hypothesis is correct, and write the answer on your paper.


So, here’s an introduction to what are cryptarithmetic puzzles, its synonyms, and a guide for how to solve them. Remember to use these tricks when you encounter these questions, and watch the video below.

References, Credits, and Links

  1. (n.d.). Cryptarithms – Basic Mathematics. Retrieved November 21, 2020, from
  2. Carson, (2020, November 5). The Scientific Method – Central Galaxy. Retrieved November 21, 2020, from

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