A MATHEMATICAL PROBLEM.

Otago Witness, Issue 1341, 11 August 1877, Page 21

A MATHEMATICAL PROBLEM.

• The following mathematical problem is said to have originated with Moses Mendelssohn :

Question — How can you prove that there mnst.be. .in the world at least two trees of the same number of leaves 1

Solution — It is certain that the number of trees in the world exceeds the greatesb number of leaves on any one tree. Call the greatest number of leaves x, and the number pf trees ar-plusy,,. and. suppose all the trees have different numbers irom i to" x. Then the tree x plus i must have a number of leaves ranging between zand er, and therefore there are two trees in the world .which have the same; number of leaves.

To make it plainer, let the greatest number of leaves on any one tree be 1,000,000, and the greatest number of trees 1,000,001 ; and suppose all the trees have different numbers of leaves — the fir ßt, having one leaf, the second two, the third three, &c ; and as no one tree can have more than 1,000,000 leaves, therefore the first tree over one million must have an equal number of leaves with one tree between 1 and 1,000,000, because it cannot have more than, 1,000,000, and as all numbers of leaves between 1 and 1,000,000 have been given away, one of these numbers must be repeated. Therefore there are at least two trees in the world which have an equal number of leaves.