Patio

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?

Time of Birth

A woman was born in a year that was a square number, lived a square number of years and died in a year that was also a square number. When was she born?

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?

Weekly Problem 42 - 2007

Stage: 3 Short Challenge Level:

We want to find angles $x^\circ$, $y^\circ$ and $z^\circ$ so that $$x^2+y^2+z^2=180\;.$$
We know that the largest angle must be smaller than $180^\circ$ and bigger than $180^\circ/3 = 60^\circ$. So the largest angle must be $169^\circ, 144^\circ, 121^\circ, 100^\circ, 81^\circ$ or $64^\circ$.

It can't be $169^\circ$ or $144^\circ$, because then the other two angles would have to add up to $11^\circ$ or $36^\circ$, respectively, which no two non-zero squares do. (Note that we want our angles to be non-zero in order to get a legitimate triangle.)

Moreover, the largest angle also can't be $121^\circ$, because then the other two angles would have to add up to $59^\circ$ and you can check that no two squares do.

If the largest angle is $100^\circ$ then the other two angles add up to $80^\circ$, so the second largest angle must be either $64^\circ$ or $49^\circ$. The latter gives $31^\circ$ as the smallest angle, which is not a square, whereas the foremost yields $36^\circ$, which is a square. So we found $$10^2+8^2+6^2 = 180\;.$$
If the largest angle is $81^\circ$ then the other two squares must add up to $99 ^\circ$, so the next largest angle must be either $81^\circ$ or $64^\circ$, giving the smallest angle as $18^\circ$ or $35^\circ$, neither of which are square numbers.

If the largest angle is $64^\circ$, then the other two square must add up to $116 ^\circ$, so the second largest angle must also be $64^\circ$, giving the smallest angle as $52^\circ$, which is not a square number.

So there is exactly one triangle with all three angles perfect squares, viz a triangle with angles $$\left(10^2,8^2,6^2\right)\;.$$

This problem is taken from the UKMT Mathematical Challenges.

View the previous week's solution
View the current weekly problem