Four Rocks

A farmer had a single rock he used to weigh out 40 pounds of feed. He lent this rock to his neighbor who broke the rock into four pieces. After the farmer investigated the four pieces he said “you actually did me a favor - with these four rocks I can now weight out all whole number weights from 1 pound to 40 pounds.” What are the weights of the four pieces? (Source: MAA Northeast Section problem competition, 2011)

What is x?

The four triangles are labeled. There are two rectangles in the figure, and every triangle side length is an integer.

What is the value of x?

An Efficient Ruler

Start with a blank 12 inch ruler that can be marked at the whole unit locations. If you place a mark 2 inches from one end, you can measure 12, 10 and 2 inches. A second mark, depending on where you put it can measure other lengths. Can you measure all lengths from 1 to 12 with as few marks as possible?