perplexus.info :: Just Math : Marble Muse

Marble Muse (Posted on 2014-06-23)

At the outset, each of the five friends Ade, Ben, Cal, Dan and Ethan had a positive integer number of marbles.

If Ade gave one-half of his marbles to Ben, and Ben gave a one third of what he then held to Cal, and Cal gave a quarter of what he then held to Dan, and Dan gave one fifth of what he then held to Ethan who then passes on a sixth of his holding to Ade - they would all have an equal number of marbles.

What is the minimum number of marbles that each of the five friends had at the beginning?

See The Solution Submitted by K Sengupta

Rating: 4.5000 (2 votes)

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re: Analytical Solution (spoiler)

Comment 2 of 2 |

(In reply to Analytical Solution (spoiler) by Steve Herman)

My earlier solution is marginally easier if I start with B's passes. It then goes like this:

Let y be the amount that everybody winds up with.

B has y after his pass, so he must have (3/2)y immediately before his pass. He passes y/2 to C.

We can solve for c (C's initial qty) in terms of y.

(c + y/2)*3/4 = y

so c = (4/3)y - y/2 = (5/6)y

c passes y/3 to D.

Similarly,

d = (5/4)y - y/3 = (11/12)y

d passes y/4 to E

e = (6/5)y - y/4 = (19/20)y

e passes y/5 to A.

So A, if a initially starts with qty a, he winds up with a/2 + y/5, which equals y.

therefore, a = (8/5)y, and A passes (4/5)y to B.

B winds up with

(b+(4/5)y)*2/3 = y

b = (3/2)y - (4/5)y = (7/10)y

y therefore is divisible by 6,12,20,5 and 10, whose LCM is 60.

so the minimum possible y is 60.

A starts with 96, passes 48, and ultimately receives 12, ending with 60.

B starts with 42, receives 48, passes 30, ending with 60.

C starts with 50, receives 30, passes 20, ending with 60.

D starts with 55, receives 20, passes 15, ending with 60.

E starts with 57, receives 15, passes 12, ending with 60.

Posted by Steve Herman on 2014-06-23 23:21:54

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