# Complex Problems

The puzzles are marked with stars (★) that show the degree of difficulty of the given puzzle.

## Eight Queens ★★★hstar("stt","hstt");

This is a commonly known chess problem.

The question: In how many ways can you arrange eight queens on a standard chessboard in such a way that none of them is attacking any other?

## Nineteen Numbers Net ★★★hstar("stt","hstt");

The figure shown below has nineteen circles that have to be filled with the numbers 1 up to (and including) 19. These numbers have to be placed in such a way that all numbers on any horizontal row and any diagonal line add up to the same sum.

Warning: there are many horizontal and diagonal lines, which have a different number of circles (3, 4, or 5), nevertheless all these sums have to be equal!

The question: How should the nineteen numbers be placed in the net?

## Cash for a Car ★★★

Thanks to Lucas Jones, we can present you the following puzzle:

A man is going to an Antique Car auction. All purchases must be paid for in cash. He goes to the bank and draws out \$25,000.

Since the man does not want to be seen carrying that much money, he places it in 15 envelopes numbered 1 through 15, in such a way that he can pay any amount up to \$25,000 without having to open any envelope. Each envelope contains the least number of bills possible of any available US currency (for example, no two tens instead of a twenty).

At the auction, he makes a successful bid of \$8322 for a car. He hands the auctioneer envelopes 2, 8, and 14. After opening the envelopes, the auctioneer finds exactly the right amount.

The question: How many ones did the auctioneer find in the envelopes?

In an alley, two ladders are placed crosswise (see the figure below). The lengths of these ladders are resp. 2 and 3 meters. They cross one another at one meter above the ground.

The question: What is the width of the alley?

## Cat & Mouse ★★★★hstar("stt","hstt");

Four white pieces (the mice) are placed on one side of a chessboard, and one black piece (the cat) is placed at the opposite side. The game is played by the following rules:

• black wins if it reaches the opposite side;
• white wins if it blocks black in such a way that black cannot make any move anymore;
• only diagonal moves (of length 1) on empty squares are allowed;
• white only moves forward;
• black can move backward and forward;
• black may make the first move, then white makes a move, and so on.

The question: Is this game computable (i.e. is it possible to decide beforehand who wins the game, no matter how hard his opponent tries to avoid this)?