 Saturday,  February 10, 2001
 M I N D  G A M E S Origin of the species

I always thought that pies were round, until someone told me that "pr2". However, that, too, turned out to be round only. In the beginning though, there were only nine digits. Tender, nimble minds used to play and devise endless number of games using these nine digits only. These were our best friends.

 9 96 964 9642 9642 9 87 875 8753 87531

The prehistoric men found that using the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 only once, they could make two numbers that multiplied to give the largest product. Biologists who researched on the origin of the species, suggested that these numbers might be 7,463 and 9,8512, but critics said there still was a pair that gave a greater product than these two. The part of the cave painting that contained the solution had been destroyed, so, no clue to it could be found for many years.

Then, one day, someone found that if the sum of two numbers was constant, the product of these increased as their difference decreased. We can prove this by letting x+y=k and x-y=d. Squaring both sides and subtracting, we get 4xy=(k2)-(d2). It became evident that larger digits must be as far to the left as possible, so, experts wrote down the following pairs in succession, applying this rule.The smaller of the two digits that were added at each step was attached to the larger number, so as to keep the difference between the two at a minimum. The final 1 must be placed after the smaller number.

 age 1 age 2 age 3 sum 1 1 36 38 1 2 18 21 1 3 12 16 1 4 9 14 1 6 6 13 2 2 9 13 2 3 6 11 3 3 4 10

When the first elections in the world were held, four persons contested in these and 5,219 votes were cast. The winner was told that he had beaten his opponents by 22, 30 and 73 votes, respectively. However, he was not told how many votes he had received.

This politician approached an expert, who told him, "Look Sir, If we add 22, 36 and 73 votes, respectively, to the losers totals, then all four candidates will have the same numbers of votes, out of a total of 5219+22+36+73=5344 votes. Also, each candidate will have 5344/4=1336 votes, including the winner, the total of whose votes has not been changed. Sir, this way, you can find out that you have received 1,336 votes."

In Indus Valley excavations, some stones were found, and on one of these, it was written, "I am a mathematician. I have three sons, but I have forgotten their ages. However, I know that the product of their ages is 36. The sum of their ages is equal to the number of windows in the building opposite my house. My oldest son has red hair."

For many years after the excavation, archaeologists could not deduce the respective ages of the sons of this mathematician who lived in the Indus Valley four millenniums ago. Research showed that the problem was meaningful only if all the ages were integers. Researchers assumed that, went ahead and used the first condition  that the product of the sons ages was 36. This gave them just 8 options as shown in the table.

Among these options was the correct solution and the researchers had no choice, but to choose. Calculations of the sums (written in the fourth column of the table) showed the only possible numbers of windows in the building opposite the mathematicians house. If the sums were 38, 21, 16, 14, 11, or 10, researchers would have been able to solve the problem immediately. They were unable to do so only because the number of windows in the building (and the sum of the ages) was 13. Because of this, they did not have a unique solution until they remembered the last line on the stone  "My oldest son has red hair." If the ages were 1, 6, and 6, there were older "sons", but not an older "son", and the ambiguity was resolved. The mathematicians sons were aged 2, 2, and 9.

Students of mathematics continue to excavate for more problems till this day. We wish them luck.