Wednesday, 15 November 2017

Answer to The Man Who Knew Infinity

At the end of Maths at: the Movies, The Man Who Knew Infinity Thomas posed two teasers to you.

A simple one to start you off.
I buy a bottle and a cork for £1.10. The bottle costs £1 more than the cork.

How much does the cork cost?

A moments thought should show that the bottle costs £1.05, whilst the cork costs only 5p. If you go it right first time well done! The answer most people tend to give if they don't pause for a second is 10p.

Now, for the more difficult question:

I live on a street with more than one house. All the houses on this street are numbered consecutively, 1, 2, 3,..., etc. Amazingly, I live in the house such that if you add up all the house numbers below me and all the house numbers above me then they come to exactly the same answer.

What is the minimum number of houses on this street and what is my house number?

The smallest answer, excluding the one house case is 8 houses on the street and I live at number 6, thus, 1+2+3+4+5=15=7+8.

There are actually an infinite number of increasing solutions to this problem. Although the solution can be found using basic algebra and a knowledge of continued fractions the details can get a bit hairy. Thus, I direct the interested reader to the following two wonderful expositions on the matter:

http://www.johnderbyshire.com/Opinions/Diaries/Puzzles/2009-06.html
http://www.angelfire.com/ak/ashoksandhya/winners2.html#PUZZANS4