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?
Sam displays cans in 3 triangular stacks. With the same number he
could make one large triangular stack or stack them all in a square
based pyramid. How many cans are there how were they arranged?
Does a graph of the triangular numbers cross a graph of the six
times table? If so, where? Will a graph of the square numbers cross
the times table too?
Many people worked on this problem, sent us
splendid 'write-ups' and invented new problems of this type which
will appear in the following months. Some people made a table of
the triangle numbers and some made use of algebra. Congratulations
to Kerry and Kayleigh from Lasswade High School; to Lyndsay,
Sheila, Helen, Suzanne, Peach, Cheryl, Peggy, Jennifer, Pen, Laura
and Emma from The Mount School York; to Ian from Cooper's Coburn
School; to Hannah from Stamford High School; and to Natalia,
Caroline, James and Katherine, from Hethersett High School,
Q: How high would the stack be? Would it be taller than you
To answer this I measured the
height of a few baked beans and spaghetti tins at home and found
that the average height was 11 cm. I then did the
[Note: Whiskas cans are 11 cm (4.25 inches) high so a stack
14 cans high is 154 cm (59.5 inches or just under 5 feet)]
Q. Felix buys 33 of these cans
and Sam stacks all the remaining cans into two identical stacks.
Find the height of the stacks.
Answers can be found by inspection using a list of triangular
Sheila shows how you can use algebra whenever you want to
find if there is triangle number of a particular size (here it is
I first found the rule for the Triangle Numbers, since that is what
we are dealing with here; it is
There is a solution which is a positive whole number, so
Q: What is the smallest number
of cans Felix could have bought leaving exactly the right number
for Sam to make two identical triangular stacks.
We need to find a number
The closest triangle number to 52.5 which satisfies this
inequality is the 9th one, 45. Therefore Sam could stack them in
two stacks with 9 rows each and the smallest number Felix could buy
to leave exactly enough for these two stacks is :
Q. Tom buys 7 cans from a stack with 9 rows. Sam re-stacks
the remaining cans into two new triangular stacks with different
numbers of rows. How many rows do the two new stacks
To find the number of rows in each stack, we use the list and
find the pair that adds up to 38 : 10 and 28, or 4 rows and 7 rows.
Q. Are there only two possible ways to arrange 49 cans into
3 triangular stacks?
No, the possibilities are:
Q. Can you find another number which can be split into 3
triangular numbers in more than one way?
There are many
solutions here. Are there infinitely many such triples of triangle
numbers? They are easy enough to find by trial and error. Can you
find a systematic method for generating them?