## Puzzle Club

Many friends have contributed to this list. The names that I remember appear interspersed or at the bottom. Some problems here are also from some books (Gardner/Smullyan ...) and columns (especially scientific american).

```(0) A says to B: I am twice as old as you were when I was as old as you are.
The sum of our ages is 63. What are our ages?

Somewhat nonsensical since B would know his age and would just have to
subtract it from 63. Nevertheless ...
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Weighing etc.
(1) There are 10 machines each making 10gm globules. One of them goes
bonkers and starts making nine gm. globules. However, you can not make it
out from its looks. You have access to a spring balance and are allowed
exactly one weighing. Find out the erring machine.

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(2) Same as above, but now more than one machine could be producing nine
gm globules. Again, find in exactly one (1) weighing the erring machines.
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(3) Closely related is a puzzle where you have a proper two pan weighing
machine. You are to come up with four (4) wts so that you can measure all
integral wts from 1 KG to 40 KG, both inclusive.
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Gautam has an interesting variation on (1) and (2)

(4) What if exactly two machines are producing 9gm globules?
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(5) Given 12 identical looking globules of which one does not weigh the same
as others, find it using a two pan machine without weights. You are allowed
three weighings. Also find if the erring globule is lighter or heavier.
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Another contibution from Gautam:

(6) A jailer goes romantic on indepence day and decides to release some of
the 100 prisoners in his jail. he decides to play a game to selct the lucky
ones. he stands in front of every cell, with number n, where n goes from 1
to 100. he divides n by all integers from 1 to n. if n is divisible by an
integer the state of the door flips, that is if it is open, it closes and
viceversa. This way if the state of the door ends up in open state then
the prisoner is set free. he does this for all the cells. Find
out which prisoners will be set free on the independence day. I hope it
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One from Douglas Hofstadter's "Fluid concepts and creative analogies":

(7) Find the next term in:
0, 1, 2, 720!
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```
I play chess reasonably well but can certainly not claim to be a good player. However, I like solving puzzles. Not the kind that are from actual games which say white to play and draw, or black to play and mate in three but the kind which craftly involve queer chess laws: improbable, but not impossible. I especially like the ones composed by Raymond Smullyan. Take a look if you like looking at chess as a formal system. The toughest one (at least for me) is the zeroeth problem. There are also sundry chess puzzles like:
```(c1) what is the maximum number of queens you can place on a chess board
so that no two of them support each other?
(c2) Is it possible to find a solution to c1 above such that no three of
them are in a straight line? (straight line as in maths: a1-b3-c5)
(c3) What is the maximum number of knights ... (this is very pleasing)
(c4) how do you make a knight go all 64 squares without going to the same
square twice?

(d1) a domino is a 2x1 rectangle. 32 such will fit on to a chess board.
remove two diametrically opposite corners of a chess board. Can you now
put 31 dominoes on to the remaining chess board?
(d2) a linear trimino is a 3x1 rectangle. You can place at most 21 such on
a chess board and one square will be left over. Which one? (it can be
shown that there is one particular square that will be left out. By
symmetry its four squares of course).
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```
A couple heard from Anil Hirani:
(8) Show that in a party there are always at least two people with the same number of friends at the party (the friends of course need not be the same).
(9) Given two identical but non-uniform strings which burn in exactly 1 hour each, measure a duration of 15 minutes.
(10) Consider the sequence of first digits of 2**n (n=0,1,2,...). Its first few elements are: 1,2,4,8,1,3,6,1,2,5,... Does 7 ever appear in it? If so what is its relative frequency compared to 8?
```Sci. Am.
(11) An urn contains 50 white and 150 black marbles. You put your hand in,
withdraw two marbles.
(a) if both are white, you leave them out and put in a new black marble,
(b) if at least one is black, leave it out and put the other one
(black or white) back in. Since each opeartion reduces the number of
marbles in the urn by one, at some point the total will reach three,
then two and then one. What is the color of that marble?

(12) A circle of radius 16 cm has 650 points thrown in at random. Show that a ring
with inner radius two and outer radius three can always be placed with its center inside
the circle such that it covers 10 or more points.

Conway:
(13) An Angel and the devil take turns on an infinite chessboard. On his turn
the devil eats one square, and on his turn the angel jumps at most 1000 in x
and 1000 in y. Can the devil trap the angel so that it has nowhere to go?
(Caution: unsolved. prize money.)
```

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###### Ashish Mahabal Last Outdated: Oct 18 2001
http://www.astro.caltech.edu/~aam/fun/puzzles.html