a)Indiana has a rectangular board 14cm x 15cm and a large supply of rectangular stamps in two sizes, 4cm x 3cm and 5cm x 2cm. She dicises to cover the board exactly with some of the stamps in any orientation, but with no gaps,cute or overlaps.She may use either kind or mixture of both.
Find all the differesnt selections of stamps Indiana can use and illustrate each with a diagram.Explain why there are noe more selections of stamps
b) Georgia has a rectangular board 11cm x 6cm,three rectangular stamps of size 4cm x 3cm and three rectangular stamps of size 5cm x 2cm.Show that she cannot cover the board exactly with these stamps in any orientation, with no gaps,cute or overlaps
a) i agree with doublez here. but to give 2 solutions you could just cover the whole board with 2x5 stamps or cover the upper part with 3x4 stamps until a small part of 2cm height and 15cm width remains and cover that with 2x5 stamps. guess there are alot more.
b) 11x6 is 66cm². the stamps are stamp A 3x4 = 12cm² and stamp B 2x5 = 10cm². so u obviously cant cover it with A stamps only. if you take 5 stamps A it doesnt fit either. trying it with less stamps of type A you will find that it might only work with 3 A stamps and 3 B stamps (in theory). since the height of the board is 6 you can only use the following combinations of stamps heightwise: 1.) 2 of type A horizontally, 2.) 3 of type B vertically, 3.) 1 of type A vertically and 1 of type B horizontally.
for 1. an area of 7x6 and 1 A stamp and 3 B stamps would remain. dont see a way to cover that.
for 2. an area of 6x6 and 3 A stamps would remain. also impossible.
for 3. ahh too difficult to describe and im too lazy to paint it. but it doesnt work either :-)
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Originally posted by bellyflopB:
a)Indiana has a rectangular board 14cm x 15cm and a large supply of rectangular stamps in two sizes, 4cm x 3cm and 5cm x 2cm. She dicises to cover the board exactly with some of the stamps in any orientation, but with no gaps,cute or overlaps.She may use either kind or mixture of both.
Find all the differesnt selections of stamps Indiana can use and illus ...[text shortened]... nnot cover the board exactly with these stamps in any orientation, with no gaps,cute or overlaps
Consider the length 6 sides of the 11*6 cm board. The combinations of card edges that can span this are (i) 3 2cm edges, ii) 2 3cm edges or iii) a 2cm edge plus a 4cm edge. We need 2 of these combinations to cover both 6 length sides.
if we try combinations i) and ii) we are left with a 6*2 cm piece of board, to cover with a 4*3 card, which doesn't work.
We can't use both combinations i) and iii) because we would need one more 2*cm stamp than we have, equally we can't use 1) twice or ii) twice because we haven't got enough stamps.
If we try ii) and iii) we find we have a 2cm wide chunk that is only 4 cm heigh: too small for any stamp.
and if we try iii) twice we can't put the two 5*2cm stamps opposite each other because an uncoverable 1cm wide piece of board is left. If altrenatively we offset the two 5*2cm stamps we are left with a zig-zag shaped bit of board which can't be covered with the remaining two stamps.
So there is no way to cover the board.