Impossible Cryptarithm

Logic Level 5

$\Large \overline{ABC} \times \overline{DE} = \overline{FG} \times \overline{HI} = X$

If $A,B,C,D,E,F,G,H,I$ are distinct positive digits, find the maximum value of $X$ .

This problem is a part of this set (click here) .

The answer is 7448.

2 solutions

Bill Bell
Aug 22, 2015

Ye olde brute force.

from itertools import permutations

maxX=0
for A,B,C,D,E,F,G,H,I in permutations([1,2,3,4,5,6,7,8,9]):
    ABC=100*A+10*B+C
    DE=10*D+E
    prod1=ABC*DE

    FG=10*F+G
    HI=10*H+I
    prod2=FG*HI

    if prod1==prod2:
        maxX=max(prod1,maxX)

print maxX

Nihar Mahajan
Jul 23, 2015

$\Large 532 \times 14 = 98 \times 76=7448$

Moderator note:

It's better to explain the steps to your answer rather than just revealing the actual numbers.

Peculiar setup. Is there a motivation behind the creation of this problem?

It's easy to see why the FG x HI = 76 x 98

First you choose the largest 2 digit composite number, then you choose the second largest 2 digit composite number that doesn't share the same digits as the previous 2 digit number. then make sure it can be converted to ABC x DE.

If it can't be satisfied, choose the 2nd largest 2 digit composite number and repeat the process.

Pi Han Goh - 5 years, 10 months ago

Please post the "solution" and not the "answer",no point.

Adarsh Kumar - 5 years, 10 months ago

I don't have a proof to show that this is maximum. If there is no point in posting the answer , I can delete it .

Nihar Mahajan - 5 years, 10 months ago

I was actually asking for,how you got this answer?