Suppose you have a 10 × 10 checkerboard and a deck of 2 × 2 cards with squares that match the size of the squares of the checkerboard, so 25 of these cards can be used to completely cover the checkerboard. If we allow the cards to overlap each other, there are many ways to cover the checkerboard. We say that an arrangement of cards is a covering if all of the cards in the arrangement are lined up with the squares on the checkerboard, and they are completely on the checkerboard, possibly overlapping, and every square of the checkerboard has at least one card on top of it. We call a covering of the checkerboard redundant if one of the cards can be removed and the checkerboard is still covered. A covering of the checkerboard is non-redundant if it is no longer a covering if any card is removed. Clearly, the smallest non-redundant covering has 25 cards. The aim of this problem is to find bounds on the number of cards in the largest possible non-redundant covering. a) Show that t
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no-redundant means some part of the larger tile is non covered
for the largest possible redundant to occur means lager part of the floor is not covered by the tiles
since the floor fits perfectly the there is no half tiles involved
hence tiles can be arranged in any part as long as one overlaps to give the largest number of non-redundance
hence out of 25 tiles the highest non redundance =24!
this gives you the number of times the cards can be arranged to attain the largest redundance
for the largest possible redundant to occur means lager part of the floor is not covered by the tiles
since the floor fits perfectly the there is no half tiles involved
hence tiles can be arranged in any part as long as one overlaps to give the largest number of non-redundance
hence out of 25 tiles the highest non redundance =24!
this gives you the number of times the cards can be arranged to attain the largest redundance
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