## Brainteaser 1675: The A-to-Z of sport

**From The Sunday Times, 16th October 1994** [link]

The individual sport or sports of the 26 members of the Venerable Sports Club means that they are just enough to form:

a football team (11); or

a hockey team (11); or

a rugby team (15).The number of members playing only two of these sports is the same as the number of members playing rugby only, which is less than one third of the total membership.

How many members play all three sports?

This puzzle was included in the book *Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1675]

## Jim Randell 9:46 am

on5 September 2019 Permalink |We can label the 7 non-empty parts of the Venn diagram, F, H, R, FH, FR, HR, FHR, according to which sports each member plays.

We can then express the constraints in terms of these 7 integer variables.

I expressed the constraints as a

MiniZincmodel, and then used theminizinc.pywrapper to collect the solutions and count them using Python.This program runs in 263ms.

Solution:2 members play all three sports.There are 7 possible solutions:

LikeLike

## Jim Randell 12:32 pm

on5 September 2019 Permalink |And here’s a solution in Python. It runs in 93ms.

Run:[ @repl.it ]LikeLike

## John Crabtree 12:41 am

on6 September 2019 Permalink |Let

Rmembers play rugby only and letAmembers play all three sports.Considering that there are 26 members and 37 players leads to

2A + R = 11,R ≤ 7and soA ≥ 2.15 players play rugby, ie

A + 2R ≥ 15, which leads to3A ≤ 7.And so 2 members play all three sports.

LikeLike