Around 1919, the British mathematician G. H. Hardy hopped on a London cab on his way to visit his Indian colleague Srinivasa Ramanujan.
The cab’s licence number, 1729, seemed dull to Hardy but his pal fervently disagreed. “It is a very interesting number,” said Ramanujan. “It is the smallest number expressible as the sum of two cubes in two different ways.”
1729 = 13 + 123 = 93 + 103
Hardy’s anecdote is one of the most well-told in British mathematics, and has given rise to taxicab number, which is any number that is the smallest number expressible as sums of two cubes in n different ways.
It is also why the UK’s first specialist maths secondary school, which opens in London in September next year, will be called the 1729 Maths School.
Today’s puzzles are loosely based on 1729, apart from the one that isn’t.
1. Square pair
What is the smallest number that can be expressed as the sum of a pair of squares in two different ways? Hint: it is less than 100
2. Strip tease
I have five strips of wood with lengths 1, 2, 7, 17 and 29 centimetres. It is impossible to arrange any three of these strips into a triangle. I would like to add another strip of wood so that I still cannot take three strips and make a triangle.
How many different lengths are possible for the seventh strip, and what are they? Lengths must be a whole number of centimetres. As a bonus, what shape can I make using only these hypothetical extra strips?
CLARIFICATION: 29cm is the maximum length. And no two lengths can be the same.
UPDATE: There was an error in this problem. Now the 9 has been deleted it works.
3. Sick sixth
I have four numbers, a, b, c, and d.
There are six ways to multiply two of these numbers together: ab, cd, ac, bd, ad, bc.
The values of five of these products, but not necessarily in this order, are 2, 3, 4, 5 and 6.
What is the value of the sixth product?
I’ll be back at 5pm UK with the answers. PLEASE NO SPOILERS. Instead discuss London taxis.
UPDATE: Read the answers here.
The UK currently has nine “university maths schools”, which are sixth form colleges focussed on maths and maths-aligned subjects. They aim to nurture top mathematical talent from underrepresented groups.
The 1729 Maths School will be the first maths school for pupils from the age of 11 (year 7). It will be based in the campus of Mill Hill School, an independent school in north London.
Its first cohorts, years 7 and 9, will have eighteen and ten pupils. The school is backed by the maths philanthropists Alex Gerko and his wife Elena, whose fortune comes from the algorithmic trading firm XTX Markets. Bursaries will be offered to pupils unable to pay the fees.
Applications are now open for years 7 and 9 (for children currently in years 6 and 8). The deadline is October 31.
For more information here’s the brochure. If you are a parent or teacher of a young mathematician with exceptional potential, please register your interest on the website and a representative from the school will reach out.
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
