You need to make a set of five rods for a magic trick, with numbers on them ranging from 1 to 31 inclusive. Then you ask people to pick a number and show you only the top number of whatever rods it is on. From that you have to guess which number the person picked. This has to be done within a few seconds and isn't a matter of remembering what numbers are on them.

What numbers should be on each of the five rods?

If by "the top number" is meant the highest number, the following set will do, albeit with a little more mental activity on the part of the magician:

rod 1:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30
rod 2:
1, 4, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29
rod 3:
1, 2, 3, 8, 9, 10, 11, 16, 17, 18, 19, 24, 25, 26, 27
rod 4:
1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23
rod 5:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

These are the numbers that have a zero, rather than a one, in their corresponding binary positions. To use them, the magician must subtract each of the top numbers from 31, then add up those differences and subtract from 31. For example, 12 is on rods whose highest number is 30, 29 and 15. Subtracting each from 31 is 1,2 and 16; add these to get 19 and subtract that from 31 to get 12.

If the person tells you it's on no rods, it's 31, as the total is zero to be subtracted from 31.

Posted by Charlie on 2003-11-11 15:40:32