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, Norfolk.

Q: How high would the stack be? Would it be taller than you
are?

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
sum

So the height of the stack would be about 154 cm. My own
height is 1 m 73 cm and the difference is 19 cm, so I would be
taller than the stack. (James)

I am 149 cm so they are taller than me. (Kerry)

It wouldn't be taller than me, I would just be able to see over the top of it (Hannah)

*[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 numbers:

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 | 66 | 78 | 91 | 105 | 120 | 136 | 153 | 171 | 190 | 210 |

*Sheila shows how you can use algebra whenever you want to
find if there is triangle number of a particular size (here it is
36).*

I first found the rule for the Triangle Numbers, since that is what
we are dealing with here; it is

$\begin{eqnarray} \\ 2 \times \frac{1}{2} \times (x^2 + x) &=& 72\\ x^2 + x - 72 &=& 0 \\ (x - 8)(x + 9) &=& 0. \end{eqnarray}$

There is a solution which is a positive whole number, so

$x = 8$ .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

$n$ such that $2 n < 105$ and $n$ is the triangle number closest to 105 when doubled. We can also say that we need to find $n < 52.5$ .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 :

$105 - (45 \times 2) = 105 - 90 = 15$.(Lyndsay)

*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
have?"*

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. (Ian)

*Q. Are there only two possible ways to arrange 49 cans into
3 triangular stacks?*

No, the possibilities are:

$\begin{eqnarray} \\ 9 \textrm{rows} + 2 \textrm{rows} + 1
\textrm{row}&:& 45 + 3 + 1 = 49 \\ 8\textrm{rows} +
4\textrm{rows} + 2\textrm{rows}&:& 36 + 10 + 3 = 49 \\ 7
\textrm{rows} + 5\textrm{rows} + 3\textrm{rows}&:& 28 + 15
+ 6 = 49." \end{eqnarray}$

(Kerry)

*Q. Can you find another number which can be split into 3
triangular numbers in more than one way?*

Kerry:

$\begin{eqnarray} 3 rows + 4 rows + 6 rows &:& 6 + 10
+ 21 = 37\\ 5rows + 1row + 6 rows&:& 15 + 1 + 21 = 37.
\end{eqnarray}$

James:

$\begin{eqnarray} 3 rows + 4 rows + 6 rows &:& 3 + 15
+ 28 = 46 \\ 3 rows + 4 rows + 6 rows &:& 10 + 15 + 21 =
46. \end{eqnarray}$

Lyndsay:

$\begin{eqnarray} 4rows + 7rows + 7rows &:& 10 + 28 +
28 = 66\\ 1rows + 4 rows + 10rows &:& 1 + 10 + 55 = 66.
\end{eqnarray}$

Nisha:

$\begin{eqnarray} 3rows + 7rows + 8rows&:& 6 + 28 + 36
= 70\\ 1rows + 2rows + 11rows &:& 1 + 3 + 66 = 70.
\end{eqnarray}$

Hannah:

$\begin{eqnarray} 3 rows + 5rows + 10rows &:& 6 + 15 +
55 = 76\\ 4rows + 6rows + 9rows &:& 10 + 21+ 45 = 76.
\end{eqnarray}$

Sheila:

$\begin{eqnarray} 2rows + 3rows + 13 rows &:& 3 + 6 +
91 = 100\\ 3 rows + 7rows + 11rows &:& 6 + 28 + 66 = 100.\\
1row + 7rows + 13rows &:& 1 + 28 + 91 = 120\\ 3 rows +
8rows + 12rows &:& 6 + 36 + 78 = 120\\
\end{eqnarray}$

Natalia and Caroline:

$\begin{eqnarray} 5rows + 7rows + 12rows &:& 15 + 28 +
78 = 121 \\ 3 rows + 4 rows + 14rows &:& 6 + 10 + 105 = 121
\\ 1row + 5rows + 14rows &:& 1 + 15 + 105 = 121.
\end{eqnarray}$

*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?*