A Simple Puzzle

[Amit N. Agarwal]

Once upon a time there was a king who had three queens. (At that time
it was politically correct.)

One day a merchant presented him with a bag full of diamonds. The king
announced that he will equally distribute them amongst his wives later.

However, the queens were of somewhat impatient nature. That night,
one queen stole one-third the number of diamonds. Later that night,
the second queen came and stole one-third the number left and as you
can guess now, the third queen also came and stole one-third the
number left. (They were all pretty honest, not stealing all).

Next morning, the king had the bag brought to him, divided into three
portions with one diamond left over. He gave one portion to each of the queens
and kept the lone diamond for himself.

What was the number of diamonds in the bag to start with?

(To be technically right, what is the smallest such number?)

[Carl J Lydick]

In article <1992Nov20.2...@EE.Stanford.EDU>, you write:
=One day a merchant presented him with a bag full of diamonds. The king
=announced that he will equally distribute them amongst his wives later.
=
=However, the queens were of somewhat impatient nature. That night,
=one queen stole one-third the number of diamonds. Later that night,
=the second queen came and stole one-third the number left and as you
=can guess now, the third queen also came and stole one-third the
=number left. (They were all pretty honest, not stealing all).
=
=Next morning, the king had the bag brought to him, divided into three
=portions with one diamond left over. He gave one portion to each of the queens
=and kept the lone diamond for himself.
=
=What was the number of diamonds in the bag to start with?
=
=(To be technically right, what is the smallest such number?)

<SPOILER>

Well, the bag ends up, in the morning, with 3n+1 diamonds, so it started out
with (3n+1)*(3/2)*(3/2)*(3/2) = (3n+1)*27/8 diamonds. So we simply need to find
the smallest n for which 3n+1 is an integral multiple of 8. That is n = 5. So
in the morning, there were 16 diamonds, meaning it started with 54 diamonds.
So:
Originally: 54 diamonds
After 1st wife: 36 diamonds
After 2nd wife: 24 diamonds
After 3rd wife: 16 diamonds
In the morning, each wife gets 5 diamonds and the king keeps one. So:
The 1st wife ends up with 23 diamonds.
The 2nd wife ends up with 17 diamonds.
The 3rd wife ends up with 13 diamonds.
The king ends up with 1 diamond.
For a total of 54 diamonds.
--------------------------------------------------------------------------------
Carl J Lydick | INTERnet: CA...@SOL1.GPS.CALTECH.EDU | NSI/HEPnet: SOL1::CARL

Disclaimer: Hey, I understand VAXen and VMS. That's what I get paid for. My
understanding of astronomy is purely at the amateur level (or below). So
unless what I'm saying is directly related to VAX/VMS, don't hold me or my
organization responsible for it. If it IS related to VAX/VMS, you can try to
hold me responsible for it, but my organization had nothing to do with it.

[Nairo Aparicio]

The answer is 54.

If x is the initial number of diamonds, the king found

8*x/27 = 3*n+1 diamonds

so n=5 is the small number such that x is an integer.

Nairo Aparicio

[Ramanendra Ravi]

How about 54 !!!!

Raman.
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[Daniel Shoham]

54.
The 1st Q takes 18 and leaves 36.
The 2nd Q takes 12 and leaves 24.
The 3rd Q takes 8 and leaves 16.
The king gives 5 to each Q, and is left with 1.

The only larger numbers that work are given by 54+81k where k=1,2,...

BTW, the Qs aren't that honest. They should have refused the next morning's
cut if they each thought they have allready gotten their share.

Dan Shoham sho...@ll.mit.edu
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[Achut Reddy]

aga...@jans.Stanford.EDU (Amit N. Agarwal) writes:

>What was the number of diamonds in the bag to start with?

>(To be technically right, what is the smallest such number?)

There is no smallest such number!

To be technically right, you mean what is the smallest such
*positive* number. Then you get the desired answer of 54.

Otherwise -27 works, e.g:

The bag initially contained -27 diamonds (*).
The 1st Q removes -9 (i.e. she added 9) leaving -18.
The 2nd Q removes -6 (i.e. she added 6) leaving -12.
The 3rd Q removes -4 (i.e. she added 4) leaving -8.
The K gave -3 to (i.e. took 3 from) each Q, leaving
1 (positive) diamond for himself.

(*) You can think of a negative diamond as a mini-black hole
which can suck up a positive diamond, and then vanish into
nothingness. (If you are less prone to flights of fancy,
you can just think of it as an IOU for 1 diamond).

-achut