# Problems of the Week

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# 2018-01-22 Basic

I have three numbers, and the product of any two of the numbers is equal to the other remaining number: \begin{aligned} ab &= c \\ bc &= a \\ ca &= b. \end{aligned} Is it true that the three numbers must be either all 0s or all 1s?

It is possible to arrange four copies of an equilateral triangle to cover an equilateral triangle with twice its side length. The same holds for a square:

However, doing the same exercise with a regular polygon with more sides requires some overlapping:

Covering a regular pentagon with double the side length requires 6 pentagons

How many identical regular hexagons are needed to cover a regular hexagon with double the side length?

If I randomly pick a card from the ten above (ace = 1), then the probability of it having an odd number or an even number is equal.

If I instead randomly pick two distinct cards from the ten above, is the probability of the two numbers having an odd sum or an even sum still equal?

A doctor soaks a cast with body-temperature water and wraps it around a patient’s arm. As the cast begins to set, does the patient feel a difference in temperature?

Note: This is a hydration reaction. As the cast hardens, water molecules "attach themselves" to the cast.

Magnus, Garry, and Fabiano are playing in an exhibition badminton event. In this exhibition, after two players face off for a set, the winner of the set stays on the court to play the player who was sitting out.

At the end of the event, Magnus has played 9 sets, Garry has played 14 sets, and Fabiano has played 15 sets.

What pair played set number 13?

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