Nerdy Breakfast

The breakfast on the hypotenuse is equal to the breakfasts on the other two sides.

Pythagorean breakfast.

Pythagorean breakfast.

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Sequences game

I made a new game for by website at http://mathewpeet.org/, you can play [here].

In the game you have to say what is missing from the sequence presented to you.

I will be grateful for any suggestions for sequences I can add.

Dividing a line

Here is a solution I found for dividing a line in 3 (or any other number of divisions) using a compass and a straight edge.

First construct a line at right angles to the line you want to divide, use the compass to make the line the same length as the first line. Then use the compass to make 2 marks along the second line, and one mark along the first line. Drawing a random circle off to the right is completely optional.
[Diagram showing second step]

Draw a line to construct a right triangle with the ratio 3:1 between it’s two sides, then construct a line perpendicular to the hypotenuse by drawing two circles centered on the line.
[Diagram showing third step]

Make another perpendicular line which is perpendicular to the previous one, but also passes through the end of second line we drew. This line will divide the original line, passing through at a third of the distance along.
[Diagram showing penultimate step]

Either repeat steps to divide the remaining part, set compass to the length of 1/3 we know have found or bisect the remaining portion of line in this case.

This a much easier problem than dividing an angle into 3 equal parts.