## Challenge 241: Baffling Boots

**a** You are faced with 12 identical pairs of Wellington boots that have been lined up in a row of 24 boots, but with left and right boots jumbled up. Show that it is possible that the boots are arranged in such a way that no set of 10 consecutive boots contains 5 pairs of boots.

**b** Now you are faced with 15 identical pairs of Wellington boots that have been lined up in a row of 30 boots, but with left and right jumbled up. Is it possible that the boots are arranged in such a way that no set of 10 consecutive boots contains 5 pairs of boots?

**c** Finally, and most challengingly, you are faced with *n* identical pairs of Wellington boots that have been lined up in a row of 2*n *boots, but with left and right jumbled up. For which values of *n* (greater than or equal to 5) is it is possible that the boots are arranged in such a way that no set of 10 consecutive boots contains 5 pairs of boots?

*Note: in all parts of this challenge any right boot may be put with any left boot to make a pair.*