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A square patio was tiled with square tiles all the same size. Some of the tiles were removed from the middle of the patio in order to make a square flower bed, but the number of the remaining tiles was still a square number. What were the dimensions of the patio and the flower bed?

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Square Routes

How many four digit square numbers are composed of even numerals? What four digit square numbers can be reversed and become the square of another number?

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What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A

Time of Birth

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

We are looking for solutions to the equation $x^2 = y^2 + z^2$ where the woman was born in the year $y$ (between $0$ A.D. and $1997$), she lived $z$ years (at least $1$ year and not more than $120$ years) and she died in the year $x$ where $x < 1997$ . There are twenty possible solutions. If she was born in $0$ A.D. there are ten possible solutions. The remaining solutions are:

Born Lived Died
$y^2$ $z^2$ $x^2$
$9$ $16$ $25$
$16$ $9$ $25$
$36$ $64$ $100$
$64$ $36$ $100$
$144$ $25$ $169$
$144$ $81$ $225$
$225$ $64$ $289$
$576$ $49$ $625$
$576$ $100$ $676$
$1600$ $81$ $1681$

This can never happen in the future (taking $x^2 > 1997$). Even if the woman had a longer life there are still no solutions in the past for a lifespan of $121$ years, but there are four solutions for a lifespan of $144$ years. In the next $2000$ years (assuming a lifespan of no more than $400$ years) the only solutions are:

Born Lived Died
$y^2$ $z^2$ $x^2$
$2304$ $196$ $2500$
$2304$ $400$ $2704$
$3600$ $121$ $3721$
$3969$ $256$ $4225$

The sets of numbers $x$, $y$ and $z$ are Pythagorean triples.