2. Whiteboardmaths.com © 2004 All rights reserved 5 7 2 1

3. GH Hardy (1877-1947) Srinivasa Ramanujan (1887-1920) ???? Number123456789101112 Cube182764125216343512729100013311728 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?) 1729 1729 = 1 3 + 12 3 and 10 3 + 9 3

4. 153 Number123456789 Cube182764125216343512729 Can you figure out what this property is? Clue 1: 1 3 = 1Clue 2: 5 3 = 125Clue 3: 3 3 = 27 1 3+ + 5 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.

5. 140141142143144145146147148149150151152153154155156157158159 160161162163164165166167168169170171172173174175176177178179 180181182183184185186187188189190191192193194195196197198199 200201202203204205206207208209210211212213214215216217218219 220221222223224225226227228229230231232233234235236237238239 240241242243244245246247248249250251252253254255256257258259 260261262263264265266267268269270271272273274275276277278279 280281282283284285286287288289290291292293294295296297298299 300301302303304305306307308309310311312313314315316317318319 320321322323324325326327328329330331332333334335336337338339 340341342343344345346347348349350351352353354355356357358359 360361362363364365366367368369370371372373374375376377378379 380381382383384385386387388389390391392393394395396397398399 400401402403404405406407408409410411412413414415416417418419 420421422423424425426427428429430431432433434435436437438439 440441442443444445446447448449450451452453454455456457458459 460461462463464465466467468469470471472473474475476477478479 480481482483484485486487488489490491492493494495496497498499 Number123456789 Cube182764125216343512729 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

6. Worksheet 140141142143144145146147148149159151152153154155156157158159 160161162163164165166167168169170171172173174175176177178179 180181182183184185186187188189190191192193194195196197198199 200201202203204205206207208209210211212213214215216217218219 220221222223224225226227228229230231232233234235236237238239 240241242243244245246247248249250251252253254255256257258259 260261262263264265266267268269270271272273274275276277278279 280281282283284285286287288289290291292293294295296297298299 300301302303304305306390308309310311312313314315316317318319 320321322323324325326327328329330331332333334335336337338339 340341342343344345346347348349350351352353354355356357358359 360361362363364365366367368369370371372373374375376377378379 380381382383384385386387388389390391392393394395396397398399 400401402403404405406407408409410411412413414415416417418419 420421422423424425426427428429430431432433434435436437438439 440441442443444445446447448449450451452453454455456457458459 460461462463464465466467468469470471472473474475476477478479 480481482483484485486487488489490491492493494495496497498499 Number123456789 Cube182764125216343512729