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Chong Ching Tong from River Valley High School, Singapore and Andrei Lazanu, age12, School: No. 205 Bucharest, Romania approached this problem in different ways.
Here is Chong's working:
Here is Andrei's solution:
First I demonstrate that 8778, 10296 and 13530 are triangular numbers, i.e. they can be written in the form n(n+1)/2. In order to do this I decomposed the product of each of the three numbers by 2 in the hope to put it in the form n(n+1)/2. I found: 8778\times2 = 2^2\times3\times 7\times 11\times 19 = 132\times133
Now, I demonstrate that the three numbers are a Pythagorean triple. The greatest number is 13530 13530^2= 183060900
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
Mr Smith and Mr Jones are two maths teachers. By asking questions, the answers to which may be right or wrong, Mr Jones is able to find the number of the house Mr Smith lives in... Or not!