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.
Investigate polygons with all the vertices on the lattice points of
a grid. For each polygon, work out the area A, the number B of
points on the boundary and the number of points (I) inside the
polygon. Can you find a formula connecting A, B and I?
We can simplify most of these equations first, before we start
giving values to all of the letters.
As $A + C = A$ we know that $C = 0$.
As $F \times D = F$ we know that $D = 1$.
$B - G = G$, therefore $B = 2G$ (add $G$ to both sides of the
$B / H = G$ therefore $B = HG$ (multiply both sides by $H$).
Since $B = HG$, and $B$ also equals $2G$,
$H$ must be $2$.
Since $B$ is twice $G$, the options are:
So $B = 6$ and $G = 3$
$E - G = F$, therefore $E - 3 = F$.
The smallest possible value for $F$ is $4$, and this means that $E$
would be $7$.
$F$ cannot be any greater than $4$ because this would mean that $E
> 7 $ and this is not allowed.
Therefore $F = 4$ and $E = 7$
Now we can solve $A + H = E$.
This is $A + 2 = 7$, so $A = 5$.
Here are the solved equations:
And all the values of $A$-$H$: