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What are cryptarithms?

In this article, Dr Yossi Elran briefly explains what cryptarithms are, and a little bit of the history from when they first appeared.


Cryptarithms or crypt-arithmetic problems are encrypted math problems, where numbers in a given mathematical expression are represented by letters or other symbols. There are two main types of cryptarythms: Hindu problems and Alphametics. In Hindu problems, every digit in a mathematical expression is concealed with the same symbol, usually an asterisk.

Surprisingly, despite the fact that once the original digits have been replaced by asterisks and they are indistinguishable, these problems can be solved. Alphametics are puzzles where each digit is replaced by a different symbol, usually a letter. Many alphametics “spell out” words, making them more attractive and entertaining. One of the most famous was invented by Henry Ernest Dudeney, a British puzzlist, in 1924: SEND + MORE = MONEY.

By substituting S=9, E=5, N=6, D=7, M=1, O=0, R=8, Y=2 the cryptarithm translates into: 9567+1085=10652. In the second half of the week, we’ll learn how this and other cryptarithms are solved.

A bit of history

Cryptarithms first appeared in the United States in 1864, but it is believed that they were invented much earlier, in Ancient China. These original cryptarithms were mainly of the “Hindu” type. In the early twentieth century, Simon Vatriquant, a Belgian mathematician pseudo-named ‘minos’, and mistakingly called Maurice Vatriquant, took them much more seriously. He published many cryptarithms in a mathematical journal called “Sphinx”, published during the first half of the twentieth century.

Maurice Kraitchick, another well-known mathematician, was the editor of this journal. The journal was dedicated to math puzzles, a branch of mathematics that is known today as recreational mathematics. Jorge Soares, among others, has a great cryptarithm site dedicated to the Sphinx.

If you are already well-versed in cryptarithms, you might want to challenge yourselves with some of the cryptarithms on his site.

© Davidson Institute of Science Education, the educational arm of the Weizmann Institute of Science
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Maths Puzzles: Cryptarithms, Symbologies and Secret Codes

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