Sam Again
Problem
Here is a collection of puzzles about Sam's shop sent in by club members to keep you busy over the Christmas and New Year holiday. Perhaps you can make up more puzzles, find formulas or find general methods.
[Nisha's Problem:] Fiona has lots of cans of spaghetti hoops, which she wants to arrange into triangular stacks ($T$-stacks). If she had 36 cans she could do one of two things :
put 8 on the bottom row, 7 on the next row, 6 on the next and so on. This stack would be 8 layers high or $T_8$;
make two stacks: one which has 5 cans on the bottom layer and so on, which would be 5 layers high ($T_5$) and the other would have 6 on the bottom layer and so be 6 layers high ($T_6$).
But she has recently had a delivery of 100 cans, so she now has 136 cans. She can now arrange her cans into 4 $T$-stacks in two different ways.
$$T_p + T_q + T_r+ T_s= 136 .$$
Can you suggest the two different ways she could do this ? (Nisha, Mount School York)
[Hannah's Problem:] How many cans does a $T_{100}$ stack have? (Hannah, Stamford High School)
[Katherine's Problem:] Sam finds he can arrange 64 cans into three $T$-stacks in two different ways. What do you think Sam's solutions were? (Katherine, Hethersett High School, Norfolk.)
Student Solutions
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.
$\begin{eqnarray} \\ T_p + T_q + T_r + T_s &=& 136 \\ T_5 + T_6 + T_9 + T_{10} &=& 15 + 21 + 45 + 55 = 136 \\ T_3 + T_4 + T_5 + T_{14} &=& 6 + 10 + 15 + 105 = 136 \\ T_3 + T_7 + T_8 + T_{11} &=& 6 + 28+ 36 + 66 = 136. \end{eqnarray}$