Memory Match Problem #3
You are setting up for a memory match game. You have to set out 50 pairs of cards (100 total) in an orderly array on the table.
However, there is more than one way to lay out the cards. If you have to use all the cards, and they can only be laid out in either a rectangular or square array, how many distinguishable array dimensions are there in total to set up the cards?
Details and Assumptions:
- The table does not in any way limit the arrangements of the cards.
- The cards are laid out face down, so the individual card faces do not affect the permutations.
- A distinguishable array means that an array which can be rotated to match another array is the same as that array. For example, a 4 by 25 array would be the same as a 25 by 4 array.
This problem is a part of the Memory Match Problem Series .
The answer is 5.
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1 solution
To find the total number of rectangular/square arrays, we really just need to find how many distinct pairs of whole numbers there are whose product equals 100. These pairs are:
And of course we could reverse the numbers in each pair to get 5 more pairs, but we are only looking for the distinct pairs. Thus, there are a total of 5 layouts for the cards.