40 Squares Min Problem with Solutions
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| The figure below depicts one possible
solution to this problem. In order to minimize the number of
matches used you want to pack the squares as tightly as possible
so that a single match can be used to form the side of two squares
whenever possible. If this were always possible then two squares
could always be constructed from only 4 matches. This is precisely
what happens when 'interior' squares (that is, squares that are
completely surrounded by other squares) are constructed. But
every square can't be an interior square. Some must be on the
border. But we can minimize that number by creating the shortest
border around the squares. The first solution below packs the
matches in a 5 x 8 rectangle and uses (r x (c + 1)) + (c x (r
+ 1)) matches where r is the number of rows and c the number
of columns. This works out to 93 matches. |
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| The
next solution packs the square into a "6 x 6 square plus
a column of 4. It again uses 93 matches. If we count the number
of matches on the border we obtain (8 + 5 + 8 + 5 = 26) for the
first solution and (7+ (4 + 1 + 2) + 6 + 6 = 26 for the second
solution. Note that many possible packing would not give the
correct solution. For example a 10 x 4 packing would not have
given a correct solution. |
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| Some
may object to these solutions because they can see larger squares
that are formed from the smaller squares packed in these matrices.
This simply points out that our minds are active enough to make
even seemingly simple and unambiguous problems "problematic." |