qts@houxa.UUCP (J.RAMMING) (02/14/86)
keywords: marbles, hacksaw
*******
Here is a puzzle I ran across several years ago; it hasn't made
the rounds lately, so I thought I would post it.
Given: an ordinary balance, and twelve marbles. Of these twelve
marbles, 11 of them have exactly the same weight. The
twelfth marble is EITHER lighter OR heavier than the other
eleven. We do not know, at the outset, which marble it
is, or whether it is lighter or heavier.
Puzzle 1: Suppose the balance spontaneously explodes after it has
been used three times. Your task is to determine, in the
three weighings available, which is the odd marble out, and
whether it is lighter or heavier than its companions.
Puzzle 2: Prove that three weighings is insufficient for the above,
when one starts off with 14 marbles.
Puzzle 3: (For twisted minds): Find the odd marble out, and figure
out whether it is lighter or heavier, starting with 13
marbles. (Might need an extra tool or two...)
J. Christopher Ramming
UUCP: decvax!bellcore!houxa!qts
WORK: (201) 949-9531
HOME: (201) 542-2079john@cisden.UUCP (John Woolley) (02/18/86)
In article <952@houxa.UUCP> qts@houxa.UUCP (J.RAMMING) writes: >Given: an ordinary balance, and twelve marbles. Of these twelve > marbles, 11 of them have exactly the same weight. The > twelfth marble is EITHER lighter OR heavier than the other > eleven. We do not know, at the outset, which marble it > is, or whether it is lighter or heavier. > >Puzzle 1: Suppose the balance spontaneously explodes after it has > been used three times. Your task is to determine, in the > three weighings available, which is the odd marble out, and > whether it is lighter or heavier than its companions. Yes, this can be done. >Puzzle 2: Prove that three weighings is insufficient for the above, > when one starts off with 14 marbles. This also is true. >Puzzle 3: (For twisted minds): Find the odd marble out, and figure > out whether it is lighter or heavier, starting with 13 > marbles. (Might need an extra tool or two...) This is not possible. With 13, you can find which one is off-weight in 3 steps, but there's 1 chance in 13 that you won't be able to tell which way off it is. (That in itself constitutes a hint.) And now a miraculously wonderful generalization. (I'll answer in a couple of weeks.) Puzzle 4: (For very twisted minds.) Determine how many marbles you can solve the puzzle for in n weighings, as a function of n. This is not easy. Puzzle 5: (Just take a guess. Answer follows from 4.) How many weighings would it take with 1000 marbles? -- Peace and Good!, Fr. John Woolley "Compared to what I have seen, all that I have written is straw." -- St. Thomas