Sam Again

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.)