Ninedigit Problem

One of the satisfactions of recreational mathematics comes from finding better solutions for problems thought to have been already solved in the best possible way. Consider the following digital problem that appears as Number 81 in Henry Ernest Dudeney's Amusements in Mathematics. (There is a Dover reprint of this 1917 book.) Nine digits (0 is excluded) are arranged in two groups. On the left a three-digit number is to be multiplied by a two-digit number. On the right both numbers have two digits each:

158 23

79 46

In each case the product is the same: 3,634. How, Dudeney asked, can the same nine digits be arranged in the same pattern to produce as large a product as possible, and a product that is identical in both cases? Dudeney's answer, which he said "is not to be found without the exercise of some judgment and patience," was

174 32

96 58

5,568 5,568

Victor Meally of Dublin County in Ireland later greatly improved on Dudeney's answer with

584 96

7,008 7,008

This remained the record until last year, when a Japanese friend of Fujimura's found an even better solution. It is believed, although it has not yet been proved, to give the highest possible product. Can the reader find it without the aid of a computer?

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