Did you solve it? That’s Mathematics!

Earlier today I set you these four lexical-numerical puzzles, inspired by Tom Lehrer’s song That’s Mathematics. Here they are again with solutions. You will also discover the highly-anticipated winning entry to the competition for most brilliant self-referential fraction.

1. Pair and share

The words ‘zero’ and ‘one’ share letters (‘e’ and ‘o’). The words ‘one’ and ‘two’ share a letter (‘o’), and the words ‘two’ and ‘three’ also share a letter (‘t’). How far do you have to count in English to find two consecutive numbers which don’t share a letter in common?

Solution: Surprisingly there are NO consecutive numbers that don’t share letters in English. (If you cant find two non-sharing consecutive numbers under ten, the chances of finding any above ten are going to be pretty low.) It would appear that English is unusual in this regard – French, Spanish and German all have non-sharing consecutive numbers: deux/trois, siete/ocho and vier/fünf.

2. Spell it out!

‘Eleven trillion’ has an interesting property. It consists of 14 letters and when written out is 11,000,000,000,000, which consists of 14 digits.

What is the lowest number to have this same property, namely that the number of letters when written as a word equals the number of digits when written in numerals?

Solution: ‘one billion’ has ten letters and 1,000,000,000 ten digits.

3. Satisfying sentence

“This sentence contains _______ letters”

Write a number in words in the blank space in the above sentence that will make the statement true.

Solution: ‘thirty six’ or ‘thirty eight’

4. Funny fractions (and win a prize)

In the phrase “two ninths”, the fraction of letters that are vowels is two ninths. Find some other fractions that have similar self-referential properties.

Solution I asked you to send in suggestions, and I promised I would send a copy of the book That’s Mathematics to the person who came up with one I liked the most.

There were many entries, and I loved the creativity of all of them. Here are a selection of the best ones.

Several people tried to find ways to capture the fact that 22/7 is the best simple approximation of pi. Of these

TWENTY TWO SEVENTHS ARE PI (7 vowels over 22 letters) from Andrew Brindle was the neatest.

Peter Taylor created this fantastic sum of fractions:

ONE HALF PLUS ONE ELEVENTH (which equals 13/22, and has 13 consonants in 22 letters)

A reader called Steve submitted THE SQUARE ROOT OF TWO THOUSAND ONE HUNDRED AND SIXTEEN, which is has 46 letters, and which describes the number 46. (Not a fraction, bit still nice.)

Fun, but a bit cumbersome is ONE HUNDRED AND THREE THREE HUNDRED AND NINTHS from Andy Knott, which has 13 vowels in 39 letters, and 103/309 = 1/3 = 13/39.

Eric Angelini (who wrote the second puzzle in today’s column) came up with ZERO POINT FOUR HUNDRED which has 8 vowels in 20 letters and is 0.400.

But my favourite came in from Vanessa Walker:


Thanks for everyone’s entries, and sorry I was not able to reply to you all personally.

Vanessa wins a copy of That’s Mathematics, a children’s book written by Chris Smith and inspired by the Tom Lehrer song of the same name.

I hope you enjoyed today’s puzzles. I’ll be back in two weeks.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

I give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.