433. Gifted Paintings. A wealthy collector had 10 valuable paintings. He wanted to make a gift to a museum, but the collector couldn't figure out how many gift options he had: after all, he could give any one painting, any two, any three paintings, and so on, he could even give all ten paintings.

The reader might think that answering this question would require a long and tedious calculation; however, I will provide a small rule that allows one to answer in all similar cases without any difficulties and thankless work.
433. Multiply 2 by itself as many times as there are pictures, and subtract 1. Thus, 2 to the tenth power is 1024. Subtracting 1, we get 1023, which is the correct answer. Suppose we have only three pictures. Then one of them can be chosen in three ways,
\footnotetext{
* It can be said that Dudeney proved a local, not a global theorem.
}
two in three ways, and three in one way, which adds up to 7 ways. But 7 is exactly \(2^{3}-1^{*}\).