Fall 2006
There is no extra credit for solving these puzzles or problems. These are meant just for fun.
Many of these are (will be) borrowed from http://www.qbyte.org/puzzles/
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A standard 8x8 chessboard can easily be covered (tiled) with non-overlapping dominoes (1x2 pieces): simply use 4 dominoes in each row.
But what if we remove two squares---one each from diagonally opposite corners of the chessboard? Can this modified chessboard be completely covered by non overlapping dominoes?
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Given 50 positive integers written in a row; something like
7 100 4 20 ........ .. ..... ... 9 11 1
Two players play the follwing game. Each player picks either the leftmost or the rightmost integer (from the remaining list). The two players take alternate turns. In the end, each player has 25 integers. The player that has larger sum of his/her integers wins.
Prove that the player playing first has a strategy so that he/she always wins or ties.
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A confused bank teller transposed the dollars and cents when he cashed a check for Ms Smith, giving her dollars instead of cents and cents instead of dollars. After buying a newspaper for 50 cents, Ms Smith noticed that she had left exactly three times as much as the original check. What was the amount of the check? (Note: 1 dollar = 100 cents.)
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Five men crash-land their airplane on a deserted island in the South Pacific. On their first day they gather as many coconuts as they can find into one big pile. They decide that, since it is getting dark, they will wait until the next day to divide the coconuts.
That night each man took a turn watching for rescue searchers while the others slept. The first watcher got bored so he decided to divide the coconuts into five equal piles. When he did this, he found he had one remaining coconut. He gave this coconut to a monkey, took one of the piles, and hid it for himself. Then he jumbled up the four other piles into one big pile again.
To cut a long story short, each of the five men ended up doing exactly the same thing. They each divided the coconuts into five equal piles and had one extra coconut left over, which they gave to the monkey. They each took one of the five piles and hid those coconuts. They each came back and jumbled up the remaining four piles into one big pile.
What is the smallest number of coconuts there could have been in the original pile?
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