Suppose we mark five points of the surface of a sphere.
Can you prove that however the points are distributed it is always possible to draw a hemisphere that include at least four of the five points in its interior or on its boundary?
This puzzle appears in many places and was included in a maths competition as recently as the early 2000s. but I think it must go back further and may have first been published by Martin Gardner, although I don't have the reference.
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Five Points of a Sphere Puzzle
Tuesday, 25 Mar 2025, 22:39
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A classic puzzle from Martin Gardner
Saturday, 1 Mar 2025, 23:46
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Edited by Richard Walker, Sunday, 2 Mar 2025, 22:44
This puzzle can be found in many places in the internet but was first published by Martin Gardner in his 'Mathematical Recreations' column in Scientific American. It goes like this (this wording is mine, not the original which I don't have to hand.)
Two mathematicians are sitting outside a cafĂŠ, and one say to the other, âI have three children, and the product of their ages is 36. Can you work out their ages?â
âNoâ, say the second mathematician, âI donât have enough informationâ.
âThen what if I tell you the sum of their ages is equal to the number of that house directly opposite?â the first mathematician replies.
âStill not enough informationâ, answers the second.
âWell what if I tell you the oldest is called Bill?â, say the first.
âAha, now I know the ages!â says the second mathematician.
1. Can YOU work out what the ages are? (Solution appears further down.)
2. Can you find a number different for 36 but which would still let the second mathematician deduce the ages if they knew the sum of the ages and the fact that the oldest child is called Bill?
PS The second triple is wrong see comments.
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