Amazing Cryptogram #1

Logic Level 5

Two same characters indicate the same number, and two different characters indicate different numbers. Every character is an integer between 0~9, while the first digits of all numbers should not be equal to 0.

Find the sum of the dividend, divisor and quotient.

Source: This is an epitaph, made by a Japanese puzzlist Tamiya Katsuya, which is dedicated to his dead friend.

This problem is a part of <Cryptarithms> series .


The answer is 2099.

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Boi (보이) by Boi (보이) , 19, South Korea

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2 solutions

Steven Yuan
Jun 24, 2017

We will rewrite the given equation into a multiplication statement to make things easier. Let 眞 = A , = A, 如 = B , B, 實 = C , = C, 生 = D , = D, 滅 = E , = E, 已 = F , = F, and 空 = G . = G. From the given problem, we can immediately find that G = 0. G = 0. (Interestingly, 空 in Chinese means "empty," so the value makes a lot of sense.) Now, we can rewrite the problem to say

A B × B C D 0 F D F E D E E F \begin{array}{cccc} & & A & B \\ & \times & B & C \\ \hline & D & 0 & F \\ D & F & E & \\ \hline D & E & E & F \end{array}

Notice that B × B B \times B ends in a digit E ≠ B . E \neq B. This means B ≠ 1 , 5 , 6. B \neq 1, 5, 6. By the same logic, C ≠ 1. C \neq 1. Also, since F > 0 , F > 0, D F E ‾ > D 0 F ‾ , \overline{DFE} > \overline{D0F}, implying that B > C . B > C. Finally, by looking at the first and second columns of the addition, we deduce that F < E , F < E, since otherwise we would be forced to carry over to the thousands place, and that is not allowed.

All that is left to do is test various combinations of B B and C C to find the one that satisfy these conditions and ensures that each letter is given a distinct value. After extensive testing, we find that the only combination that works is ( B , C ) = ( 7 , 4 ) , (B, C) = (7, 4), yielding the solution ( A , B , C , D , E , F , G ) = ( 2 , 7 , 4 , 1 , 9 , 8 , 0 ) . (A, B, C, D, E, F, G) = (2, 7, 4, 1, 9, 8, 0). Thus, the sum of the dividend, divisor, and quotient is equal to D E E F ‾ + A B ‾ + B C ‾ = 1998 + 27 + 74 = 2099 . \overline{DEEF} + \overline{AB} + \overline{BC} = 1998 + 27 + 74 = \boxed{2099}.

The completed cryptarithm is

2 7 × 7 4 1 0 8 1 8 9 1 9 9 8 \begin{array}{cccc} & & 2 & 7 \\ & \times & 7 & 4 \\ \hline & 1 & 0 & 8 \\ 1 & 8 & 9 & \\ \hline 1 & 9 & 9 & 8 \end{array}

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Interesting facts about this epitaph!

  • 74 in Japanese can be pronounced as "nanashi", which can also mean "anonymous". Since 空=0 and 空 means "nothing" as well, looking at the original division shows that you start with anonymous and end with nothing!

  • In Buddhism, the number 108 symbolizes "all feelings of the mortals", and see that the division subtracted 108. It symbolizes death itself.

  • Finally, and most surprisingly, the dividend 1998 is the year his friend died.

I got deeply impressed when I first saw the explanations on this division, so here I am, showing them to you guys!

Boi (보이) - 3 years, 11 months ago

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