These alphabet bricks are painted in a special way. A is on one
brick, B on two bricks, and so on. How many bricks will be painted
by the time they have got to other letters of the alphabet?
I have forgotten the number of the combination of the lock on my
briefcase. I did have a method for remembering it...
Sam sets up displays of cat food in his shop in triangular stacks.
If Felix buys some, then how can Sam arrange the remaining cans in
Scott Lewington, age 11, Lea Manor High School
and Ling Xiang Ning, Allan, age 12, Raffles Institution, Singapore
solved the Man Food problem.
They found square based pyramids with the same number of cans as
one triangular stack with a depth of one can (a triangular number
of cans) and then three triangular stacks to make up of the same
number of cans altogether.
The triangular numbers are: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55,
66, 78, 91, 105,...
The square based pyramid numbers are: 1, 5, 14, 30, 55, 91, 140,
As we can see, there are two numbers that appear in both lists,
55 and 91. This means that both of them are possible to be a square
pyramid (with a 5 by 5 base or a 6 by 6 base) and a triangular
stack (with a base of 10 or of 13). Now, we just have to find the
three triangular stacks 55 and 91 cans can be made into.
For 55 cans, it could be made into triangular stacks of 6, 21
and 28 (T 3 + T 6 + T 7 ).
For 91 cans, it could be made into triangular stacks of 10, 15
and 66 (T 4 + T 5 + T 11 ) or of
3, 10 and 78 (T 2 + T 4 + T 12 )
or of 15, 21 and 55 (T 5 + T 6 + T
10 ) or 10, 36 and 45 (T 4 +T 8 +
T 9 ).
There are other possible solutions. This type
of investigation can best be pursued using a spreadsheet with
formulae for the different types of patterned numbers in different
columns. This leads to conjectures about the algebraic
relationships between different patterned numbers which may be