What are the missing numbers in the pyramids?
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number.
Cross out the numbers on the same row and column. Repeat this
process. Add up you four numbers. Why do they always add up to 34?
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.
The following solution was sent in by Ian Wasson (Form 3),
Ballymena Academy and we have included a diagram sent by Kang Hong
Joo of the Chinese High School, Singapore. Well done also Sarah
Meredith, class 1M, Madras College.
To have all green you must first balance the yellow and brown.
To do this you can lose one yellow and gain two brown by having a
yellow meeting a green chameleon to produce two brown
If each brown meets a yellow chameleon you will eventually end
up with them all being green. Seventeen meetings will have occurred
here plus the one which occurred early giving a total of eighteen
meetings necessary to make all the chameleons green.
After 1st meeting:
After 18th meeting:
Note that after each meeting the numbers of all three colours
have the same remainder when divided by 3. (Using the language of
modulus (or clock) arithmetic they are said to be equivalent modulo
(2) The second part was explained by a student from St Peters'
The only way to make all the chameleons the same colour is to
have a difference between two numbers which is a multiple of three
(i.e. 3, 6, 9, 12, 15, 18...) which means two numbers must have the
same remainder when divided by three. With 13, 15, 17, the
remainders are 1, 0, 2. No matter which two colours meet, the
remainders will be 0, 2, 1. The next time two meet, the remainders
will be 2, 0, 1. The next meeting, the remainders will be 1, 0, 2
and we are back to the beginning. They will always have different
remainders so it is impossible for all to be the same colour
because this can only happen when the numbers of two of the colours
are reduced to zero, in which case they have the same remainder on
division by 3.