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The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 � 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers? ### Special Sums and Products

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

# Do Unto Caesar

##### Age 11 to 14 Challenge Level:

At Caesar's Palace in Vegas, were three poker players - Alan, Bernie and Craig. At the beginning of the night the amount of money each had was in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of the players won 1 200. Suppose at the outset they had a, b and c.

This solution was sent to you by Prav Idaikkadar and Sheli Levenson (YR9) of the North London Collegiate School Maths Puzzle Club. Have a nice day

a : b: c = 7 : 6 : 5 = 7/18 : 6/18 : 5/18
6 : 5 : 4 = 6/15 : 5/15 : 4/15

The lowest common multiple of 15 and 18 is 90.

This is 35/90 : 30/90 : 25/90

36/90 : 30/90 : 24/90

We can see that Craig lost 1/90 of his money and Alan gained it. We are told that the sum of money gained is 1,200. Therefore 1/90 of the money = 1,200.

The total amount of money is 90 * 1,200 = 108,000.

In the beginning of the game Alan had 35/90 of the total money

35/90 * 108, 000 =  42, 000.

Bernie had 30/90 of the total money and this is

30/90 * 108, 000 = 36, 000.

Craig had 25/90 of the total money and this is

25/90 * 108, 000 = 30, 000.

These were the assets of the players at the beginning of the evening

Well done Prav and Sheli. Other very good solutions were sent in by Elizabeth Wells, also from the N.C.L.S. puzzle club and James Page from Hethersett High School, Norwich.