Enigma 1720: Tables seating four
17 October 2012
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From New Scientist #2887, 20th October 2012 [link]
Russian peasants were said to perform multiplication knowing only how to add, and to multiply and divide by 2. I showed my niece how to multiply 25 by 31 by this method, as shown in the first box.
In the left column, the numbers are divided by 2 successively, any resulting halves being ignored, until 1 is reached. The right column consists of an equal number of values, each after the initial 31 being double the previous one. A simple rule, which I left my niece to discover, tells us which of the values in the right column we must add to get the required total of 775.
My brother-in-law then sent me the following problem, which uses the above method to multiply p by P, and which, with the relevant addition, gives the correct result, ENIGMA, where each of these six letters represents a different non-zero digit. He has inserted a 4 wherever it occurs in the calculation, leaving all other place-holders unspecified, as shown in the second box.
What is ENIGMA?
Just to be clear: The left column starts with p (lower case), and the right column starts with P (upper case).