May 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 they eagerly split the jewels in the bag three ways and finding one extra, they agreed to discard it.

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 79.
The first thief took 33 jewels, the second thief took 24 jewels and the third thief took 18 jewels, with 4 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
Let E = number of jewels in the morning when one thief took his share

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)
E = (D - 1) - (D - 1)/3 = 2/3 (D - 1)

Starting with the fourth equation and working back:

3/2 * E = D - 1
3/2 * E + 1 = D

3/2 * E + 1 = 2/3 * (C - 1)
9/4 * E + 3/2 = C - 1
9/4 * E + 5/2 = C

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

27/8 * E + 19/4 = 2/3 * (A - 1)
81/16 * E + 57/8 = A - 1
81/16 * E + 65/8 = A

So, I rewrote the last equation to be:
A = (81 * E + 130) / 16

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

If E = 14, then A = 79
If E = 30, then A = 160
If E = 46, then A = 241
If E = 62, then A = 322
etc.

James Alarie sent in the following analysis:

There is an infinite number of solutions.

For any positive integer I, the total (T) number of jewels stolen will be:
T = 81 * I - 2
and the three theives A, B, and C will, on their first pass, get:
A = 27 * I - 1
B = 18 * I - 1
C = 12 * I - 1
The next morning, they will divide the remaining jewels and each receive:
E = 8 * I - 1
The totals for each will be:
A = (27 * I - 1) + (8 * I - 1) = 35 * I - 2
B = (18 * I - 1) + (8 * I - 1) = 26 * I - 2
C = (12 * I - 1) + (8 * I - 1) = 20 * I - 2
and the grand total, including the discarded 4 jewels, will be:
(35 * I - 2) + (26 * I - 2) + (20 * I - 2) + 4 = 81 * I - 2

The minimum solution is with I = 1:
T = 79
A = 33
B = 24
C = 18

Correctly solved by:

1. Anna Vice and Eliza Sheffield	Tuscaloosa, Alabama
2. James Alarie	Flint, Michigan
3. Tyler Stoddard	Mountain View High School, Mountain View, Wyoming
4. Ivy Joseph	Pune, Maharashtra, India
5. Cooper Martin	Mountain View High School, Mountain View, Wyoming
6. Brijesh Dave	Mumbai City, Maharashtra, India
7. Sameha Haque	Delta High School, Delta, Colorado
8. Trevon Peterson	Mountain View High School, Mountain View, Wyoming
9. Behlee Aimone	Mountain View High School, Mountain View, Wyoming
10. Linzy Carpenter	----------
11. Valerie Walker	----------
12. Sabre Williams	----------
13. Karlee Hereford	Mountain View High School, Mountain View, Wyoming
13. Brianna Tims	Mountain View High School, Mountain View, Wyoming
14. Jameson Price's Math Class	Springfield Catholic High School, Springfield, Missouri
15. Dawson Case	Mountain View High School, Mountain View, Wyoming
16. Nate Moretti	Mountain View High School, Mountain View, Wyoming
17. Baylee Tims	Mountain View High School, Mountain View, Wyoming
18. Kelsey Giorgis	Mountain View High School, Mountain View, Wyoming
19. Sami Lane	Delta High School, Delta, Colorado
20. Tom Moschner	Pasco County Schools New Port Richey, Florida
21. Harlan Benedict	Mountain View High School, Mountain View, Wyoming
22. Jason Stoddard	Mountain View High School, Mountain View, Wyoming

Send any comments or questions to: David Pleacher