Whiteboardmaths.com © 2004 All rights reserved

GH Hardy ( ) Srinivasa Ramanujan ( ) ???? Number Cube Hardy’s Taxi Number There is a famous story in mathematics concerning the great English mathematician G. H. Hardy and the self taught Indian genius Ramanujan. Hardy invited Ramanujan to Cambridge during the First World War to help him fill in the “gaps” in his mathematical knowledge, in particular to introduce him to the idea of formal mathematical proof. Unfortunately the food and climate did not agree with him and he became ill and spent several periods in hospital. One day, Hardy took a taxi from Cambridge to visit Ramanujan in Putney hospital. At his bedside Ramanujan asked him what the taxi number was. Hardy replied that it was a “rather dull one.” The number was ****? Ramanujan perked up and replied by saying “No, it is in fact an interesting number; it is the smallest number that can be expressed as the sum of two cubes in two different ways.” (What was it?) = and

153 Number Cube Can you figure out what this property is? Clue 1: 1 3 = 1Clue 2: 5 3 = 125Clue 3: 3 3 = = 153 Out of the infinity of numbers there are only 3 other numbers that share this property. Luckily, all of them are below 500. Can you find them? In Hardy’s book “A Mathematician’s Apology”, Hardy discusses what it is that makes a great mathematical theorem great. He chooses two that have stood the test of time: namely Euclid’s proof of the infinity of the primes and Pythagoras’s proof that  2 is irrational. He contrasts these with mere mathematical curiosities. One of the curiosities he mentions is the number 153. It has an usual property relating to cube numbers.

Number Cube Use the table of cubes below to help you find the other 3 numbers with the same property as 153. Very few calculations should have to be made if you think logically. Now solve a similar problem for square numbers. Find all two digit numbers (xy), < 100, such that x 2 + y 2 = xy

Worksheet Number Cube