Is there a square made up of square tiles that has only 75 squares overall? what about one that has 120 squares overall? if it is possible show the square design and explain how u found it. if it is not possible explain why that design cannot exist

See below.

Step-by-step explanation:

There are two essential pieces of information missing in the question, namely:

(1) "can the the square tiles be of varying size?"

(2) "what are the number constraints on the size of a tile?" (integers only? fractional lengths? Any real number?)

Without knowing the above, I can only speculate:

If the answer is "no, the square tiles are all same size, and must be of integer size" then the answer to your question will be "no, neither with 75 nor with 120 tiles a square of integer size can be covered" This is because the numbers 75 and 120 are not perfect squares (unlike, say, 121), which is easy to check.

If the answer is "yes, the square tiles can vary in size, but must be integers" then the answer to your question will be "yes" at least in the case of the 75 tiles...I could come up with an arrangement to solve that and am positive there is one for 120. I am not going into the detail here not knowing what is really required.

If the answer is "tile size can be any real number" then the answer to your question will be "yes" (albeit, there may never be possible to construct).

Let me know if you have questions.

dnji

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