#### You may also like ### Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares? ### DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}. ### Sixational

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

# Chocolate 2010

##### Age 14 to 16 Challenge Level:

There were a significant number of correct solutions to this problem and one of the most well expressed is given below. A number of younger members sent examples of the "trick" working and it was good to see you investigating what was happening. The solution chosen tries to explain why. Some of you needed to dig a little deeper! Well done to all of you who successfully solved this.

David of Colyton Grammar School gave the most comprehensive and well explained solution. Two other good solutions were received from Charlene of Sekolah Menengah Pengiran Jaya Negara Pengiran Haji Abu Bakar and Vu of School B6.

We take the number of pieces of chocolate that we can have as x.

First, we multiply it by 2, so we have 2x.

Adding 5 to it, we now have 2x + 5

Multiplying it by 50 we have 50*(2x + 5) = 100x + 250

Finally adding 1753 or 1754 to the equation on the previous line, we get 100x + 250 + 1753 or 100x + 250 + 1754, which, after subtracting year of birth (take it as y) is either

1. 100x + 2003 - y
OR
2. 100x + 2004 - y

Multiples of 100 always end in double-zero digits, so what the tens and units digits are depends on the value of 2003 - y or 2004 - y (this depends on the year you last celebrated your birthday), which gives your age! Note that the trick does not work for centenarians as their age would exceed two digits.

We also see that the hundreds digit is equal to x, provided that x is not larger than 10.

To start with you are asked to choose an integer between 1-10 which I will call "x".

WE HAVE: x
The first calculation is to multiply by 2.
SO WE HAVE: 2x
Then we are told to add 5.
SO WE HAVE: 2x + 5
Following that is an instruction to multiply by 50.
GIVES US: 100x + 250