June 2017
Problem of the Month

The Three Thieves
contributed by Jim Hooker

Three thieves stole a bag of jewels from a Jewelry store late one evening.   That night they decided to split their ill-gotten cache the next morning for obvious reasons.

During the night the first thief woke up and took the bag of jewels and split it three ways with one jewel left over, he took a third of the split and discarded the extra jewel.   He returned the remainder of the jewels to the bag and went back to sleep.

Some time later the second thief woke up, took the bag, split the jewels three ways and had one extra with which he discarded.   He took a third of the split and returned the remainder to the bag, then went to sleep.

Just a bit later the third thief woke up and finding the others sound asleep, took the bag and split it three ways and had one extra jewel.   He took a third, discarded the one jewel and returned the rest to the bag.

The next morning, after they all woke up, they divided the remaining jewels into three equal shares.   This time no jewels were left over.

Happily they went their separate ways.

How many jewels were stolen?   How many jewels did each thief leave with?

Solution to the Problem:

The smallest number of jewels that were stolen would be 25.
The first thief took 10 jewels, the second thief took 7 jewels and the third thief took 5 jewels, with 3 jewels discarded.

Let A = number of jewels in the original pile
Let B = number of jewels after the first thief has helped himself
Let C = number of jewels after the second thief has helped himself
Let D = number of jewels after the third thief has helped himself

Then you can write the following equations:

B = (A - 1) - (A - 1)/3 = 2/3 (A - 1)
C = (B - 1) - (B - 1)/3 = 2/3 (B - 1)
D = (C - 1) - (C - 1)/3 = 2/3 (C - 1)

Starting with the third equation and working back:
3/2 * D = C - 1
3/2 * D + 1 = C

3/2 * D + 1 = 2/3 * (B - 1)
9/4 * D + 3/2 = B - 1
9/4 * D + 5/2 = B

9/4 * D + 5/2 = 2/3 * (A - 1)
27/8 * D + 15/4 = A - 1
27/8 * D + 19/4 = A

So, I rewrote the last equation to be:
A = (27 * D + 38) / 8

Substitute in values of D until you get an integer value of A:

If D = 6, Then A = 25
If D = 14, then A = 52
If D = 22, then A 79
If D = 30, then A = 106
If D = 38, then A = 133
If D = 46, then A = 160
If D = 54, then A = 187
etc.

Correctly solved by:

1. Anna Vice and Eliza Sheffield	Tuscaloosa, Alabama
2. James Alarie	Flint, Michigan
3. Brijesh Dave	Mumbai City, Maharashtra, India
4. Dhanshri Patel	Ahmedabad, Gujarat, India

Send any comments or questions to: David Pleacher