Cecile asked:

‘If there are more trees in the world than there are leaves on any one tree, and no trees have no leaves, then there are at least two trees with the same number of leaves.’

Is this statement true or false. Explain in one sentence.

Answer by Craig Skinner

It’s true.

One-sentence explanation:

If every tree has a different, whole-number of leaves, then the trees can be listed in order of leaf number, and there is a one-to-one correspondence between tree number and leaf number so that the number of trees always equals the number of leaves on the leafiest tree.


Tree 1 has one leaf
Tree 2 has two leaves

Tree x has x leaves

Number of trees(x) always equals number of leaves (x) on leafiest tree. To get more trees than leaves, we need at least one tree with no leaves, or at least two trees with the same number of leaves.

I’m not sure whether such puzzles aid philosophical thinking, but they are entertaining. A compendium of maths puzzles, factoids, formulas and more is

Pickover C A (2005) A Passion for Mathematics: numbers, puzzles, madness, religion, and the meaning of life Wiley, New Jersey