The Murderous Maths Research Lab proudly presents...


Here's the question we asked in our research lab:


You have a sheet of 12 stamps measuring 4x3. How many ways can you tear out 4 stamps that are all attached to each other?

ALL rotations, translations and reflections count.

Here are four ways to get you started!

Can you make a formula that gives an answer for any size of block of stamps to be torn out from any size of rectangular sheet of stamps? E.g. the formula would work out how many ways you could tear a block of 6 stamps from a sheet measuring 8x9.

We all agreed that the first answer was that there were 65 ways to tear a block of 4 stamps from the 4x3 sheet. However, coming up with a formula was something else until we got Matthew's answer...

     a                                                                            a   

( å   S(Z(x-c-1)(y-d-1)   ) + (  å   S(Z(x-c-1)(y-d-1) + (y-c-1)(x-d-1) )

  c = d ----------------------------------------------------------            c ≠ d    ---------------------------------------------------------------------------------------------  

                              S                                                         S

And here is Matthew's explanation!

Above is my formula for the number of times p(n) (the polyominoes of n) fit in grid x by y. The Sses cancel each other out, but I included them so the sigma fuction would be satisfied.

Let n be a positive real integer. Let a be the number of polyominoes of n. Take polyomino p(s). Place it in rectangle q(s), q(s) being the smallest rectangular grid possible p(s) can fit in. q(s) has dimentions c by d. The maximum number of different ways p(s) can be placed in q(s) (when c = d) is 8, the minimum 1, and when c does not = d, the maximum is 4 times, and minimum 1. Call the number of times p(s) fits in q(s) Z.

Take rectangle R. The number of times q(s) fits in it (when c = d) is (x-c-1)(y-d-1), and when c does not = d, (x-c-1)(y-d-1) + (y-c-1)(x-d-1). Multiply this equation by Z, and that is how many different places p(s) can be placed in R. Sum all the other polyominoes with it (using the sigma function) and there is your answer (see above). So for 4 connected stamps to fit in a 4 by 3 grid would be:

(8(3-2)(4-1) + (4-2)(3-1)) + (4-1)(3-1) + (x-0)(y-3)(y-0)(x-3)

which can be simplfied as -28y + x(19y-28) + 33 which comes out as 65.

Incidentally, the number of different ways a block of 6 connected stamps can fit in a sheet 8 by 9 is 7755.

There's just one more amazing thing you should know - Matthew is aged 14!

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