I have forgotten the number of the combination of the lock on my
briefcase. I did have a method for remembering it...
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?
Here is a collection of puzzles about Sam's shop sent in by club
members. Perhaps you can make up more puzzles, find formulas or
find general methods.
A student from Mearns Castle made the
useful connection with another problem published this
Aswaath from the Garden International
School in Kuala Lumpur, Malaysia mentioned that the method for
solving this problem connected with the
method used by Gauss when he was still
a young student.
Joe from Hove Park Lower School also
noticed connections with other work:
If, for example, 10 mathematicians met, the first will make 9
handshakes, the second makes 8, the third makes 7 and so on until
the tenth finds he has already made handshakes with everyone and so
makes no more.
This gives 9+8+7+6+5+4+3+2+1+0 handshakes and this is 45.
But look at the sequence... it is the 9th triangular
Triangle Numbers and/or Clever Carl)
The formula for the Tth triangular number is T(T+1)/2
With the handshake problem, if there are n people, then the number
of handshakes is equivalent to the (n-1)th triangular number.
Subsituting T = n-1 in the formula for triangular numbers, we can
deduce a formula for the number of handshakes between n
Number of handshakes = (n-1)(n)/2
Jayme from the Garden International
School agreed and used this insight to correct Sam's
Joseph from Bradon Forest School and
Tabitha from The Norwood School used similar reasoning:
We received many more correct solutions,
including very clear ones from Siddhartha and Tasuku
from the Garden International School in Kuala
Lumpur, Ben from Bedminster Down School,
Abhinav from Bangkok Patana School and Luke from Maidstone Grammar
School. Well done and thank you to you all.