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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 2n 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.

Submit your solution

Please do send in your solution to this problem to  weeklymaths@kcl.ac.uk  You can scan or photograph your written work, or type your solutions. If this is your first weekly maths challenge solution, please include your year group and the name of the school you attend. We'll be happy to provide you with feedback on your solution.

 

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