Grilled Einstein
Aditya Rishi

Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore. — Albert Einstein (1879-1955)

PROFESSOR EINSTEIN said he would think of a two-digit integer (any integer between 10 and 99) and I would try to guess it. If the number I name is correct, or if one of its digits is equal to the corresponding digit of EINSTEIN's number and the other digit differs by one from the corresponding digit of EINSTEIN 's number, then, EINSTEIN says "hot"; otherwise, he says "cold" (If EINSTEIN 's number was 65, then, by naming any 64, 65, 66, 55 or 75, THE DETECTIVE will be answered "hot"; otherwise, he will be answered "cold"). Professor EINSTEIN asked me if there was a 22-attempt winning strategy for me?

The detective had no answer. Let's form a table in which rows represent first digit and columns represent second digit. We cover the table by 22 figures, which represent detective's moves. These have 9 crosses, 9 half-crosses, 2 dashes and 2 dots. One square (number 50) is not covered.

Here's how the detective can win: he first tries numbers that are inside crosses. If all answers were "cold", he tries numbers that are inside half-crosses. If all answers were "cold" again, he tries numbers inside dashes. If the answers were "cold", he tries numbers that are dots. If all answers were "cold" still, he knows that Einstein's number was 50.

However, at any point, Einstein may say "hot", in which case, the detective gives a numbers inside a cross, say 23. Then, in three more attempts, the detective can find out Einstein's number. If 22 is "cold" and 24 is "hot", then, Einstein's number is 24. If 22 is "hot" and 24 is "cold", then, Einstein's number is 22. If both are "cold", then, if 13 is "cold", Einstein's number is 33; otherwise, it is 13. If both are "hot", the number is 23.

If the answer is "hot" when the detective says a number inside a half-cross or a dash, then, as earlier, we can show that the detective can find out Einstein's number in two more attempts. There is a chance that the answer is "hot" when the detective tries the number inside the last dash (which would be the 20th attempt). This way, in exactly two more attempts, he will find out Einstein's number.

"Finding a solution to the problem required a tedious hit and trial with a little logic. Your problem can be split into a grid of 9 columns and 10 rows that start with 10 and goes down. Second column starts with 20 and so on. If you select a number from this grid, numbers one on left, one of right, one above, and one below besides the number itself are selected. We are to find a combination of 22 numbers that covers the whole grid. And that combination is: 10, 14, 17, 22, 27, 29, 30, 35, 43, 48, 51, 56, 64, 69, 70, 72, 77, 85, 89, 91, 93 and 97," says Dr Tarsem Lal of Khanna. Well 9 columns and 10 rows, or 10 columns and 9 rows, what difference does it make, speaking relatively. Write at The Tribune or adityarishi99@yahoo.co.in.