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## 'Sam Again' printed from http://nrich.maths.org/

You can make good use of a spreadsheet or of algebra to solve this
problem. As well as solutions to Hannah's problem, solutions are
invited to Santa's problem using methods which work equally as well
for large numbers as for 21.

Congratulations to Jamesof Hethersett High School,
Norfolk and to Helen, Charis, Lyndsay, Christiane, Charlotte,
Bellaand Rachel, Year 10, Mount School York. Some said
that either the reindeer were extremely hungry or there are more
reindeer in Santa's employ than we are led to believe (someone did
suggest 'Reindeer Express' along the lines of the Pony Express of
history!) Here is Helen's solution:

To find how many layers high the pyramid is:

$$231 {\rm (cm)} / 11 {\rm (cm)} = 21 {\rm \ layers\ high.}$$

Each layer is a consecutive square number and the square numbers
from $1^2$ to $21^2$ are:

1, 4, 9, 16. 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256,
289, 324, 361, 400, 441

All these added together gives 3311.

To find out how many cans in the triangular based pyramid, you add
up the triangle numbers from the 1st to the 21st:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136,
153, 171, 190, 210, 231

This equals 1771 cans.

The reindeer ate 3311 - 1771 = 1540 cans in one meal. There are
1771 cans left so there are enough to feed the reindeer another
meal with 231 cans remaining.

James also solved Katherine's problem as follows. He wrote down all
the triangular numbers smaller than 64 and then, using trial and
improvement, James found two solutions:

$\begin{eqnarray} \\ T_5 + T_6 + T_7 &=& 15 + 21 + 28 = 64
\\ T_2 + T_3 + T_{10} &=& 3 + 6 + 55 = 64. \end{eqnarray}$

Well done Jacqui, Year 8, Mount School York for your solution to
Nisha's problem. There are at least three ways of arranging 136
cans into four $T-$ stacks.

$\begin{eqnarray} \\ T_p + T_q + T_r + T_s &=& 136 \\ T_5 +
T_6 + T_9 + T_{10} &=& 15 + 21 + 45 + 55 = 136 \\ T_3 + T_4
+ T_5 + T_{14} &=& 6 + 10 + 15 + 105 = 136 \\ T_3 + T_7 +
T_8 + T_{11} &=& 6 + 28+ 36 + 66 = 136. \end{eqnarray}$