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A Collection of Analytical Puzzles
© Doug Peterson, 2009
Feel free to post a tough puzzle in the guestbook above!
1. In this unusual game of chess the players were permitted to arrange their eight high-ranked pieces behind the pawns in any order with the bishops on opposite colors. In addition to the normal rules for claiming a draw, a player can also declare a draw at the time his king occupies the square where it began the game if it has already occupied the square where the opposing king began the game. White's king began the game on d1 and black's king began on b8 (white's bishop is on e1). White's king has not already occupied b8. It's White's move. How does he force a sure draw?
2. The shuttle problem.
Imagine a space shuttle is returning to Earth at distance R from the center of the Earth. It is traveling with a velocity (V') in a direction towards the center of the Earth with its engines off. The shuttle is on a direct path towards the center of the Earth.
Three equations are relevant.
1) The period
of revolution of a satellite orbiting Earth is given by this equation:
Pa = 2πa3/2 / (GM)½
a is the length of the semi-major axis of the ellipse that describes the orbit of the satellite (Earth is located at one of the foci's of the ellipse)
G is the gravitational constant
M is the mass of Earth
2) The orbital velocity of the satellite is given by this equation:
v = [(2GM/r) - (GM/a)] ½
r is the distance of the satellite from the center of Earth
3) The shuttle undergoes a non-uniform acceleration on its return to Earth because the acceleration due to gravity increases. Here is the basic physics equation:
a. gr2 = GM
g is acceleration due to gravity at distance r from center of Earth with mass M
G is the gravitational constant
Can you find an equation that approximately determines how long it will take the shuttle to travel any distance (from R to r) on its return? A good approximation is sufficient.
3. Letter Cross Out
Cross out ten letters in such a way that the remaining letters spell a single word:
ASNQINNEGLLEEWTTORERD
4. The three variable problem
Find a solution for this equation:
b/(a + b) + a + b + c = b/(a + b) × a × b × c
5. The Tournament Bracket Puzzle
Eight people participated in an NCAA basketball pool in which each person picked eight teams (totaling all of the 64 bracketed teams). Each person noticed that the sum total for the seed numbers of his eight teams was 68. Each person had two teams in each of the four regional brackets with one team in the upper half of the bracket and one in the lower. Seed numbers 1, 4, 5, 8, 9, 12, 13, and 16 are in the upper half; seed numbers 2, 3, 6, 7, 10, 11, 14, and 15 are in the lower. Moreover, each person noticed that at least one of his teams played a team of each of the other seven players in the first round. A #1 plays a #16, #2 plays a #15, etc. How were the 64 teams selected? Note: this problem is difficult and only one solution might exist.
6. The Average Salary Puzzle
At dinner five men determined the average annual salary of the five using a method. After dinner two of them moved
to a distant country-- never to be heard from again. With their method no man could possibly determine the salary of any of
the other men even by subsequently conspiring with any of the other four. Moreover, no man will directly or indirectly
reveal his salary to any of the other four. One of them had no salary. How did they do it?
7. 21 Integers
I'm thinking of a sequence of 21 consecutive integers. The sum of the squares of the first 11 integers equals the sum of the squares of the last 10 integers. What is the first number in the sequence?
It's easier than you think.
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