Numberplay

The Navajo Nation Sheep Problem

By Gary Antonick
Navajo rugs at the Hubbell Trading Post in Ganado, Ariz. Jill Torrance for The New York Times
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This week’s problem was suggested by Dave Auckly, a professor of mathematics at Kansas State University. Dr. Auckly found the basic idea for the puzzle in the fraction section of Gordon Hamilton’s Math Pickle website while collecting problems and activities for the Navajo Nation Math Circle Project. Let’s give it a try. Here’s —

The Navajo Sheep Problem

The Navajo people are known for weaving some of the most delicate and sophisticated textiles in the world, working with wool from local sheep. One day a sheep trader arrives at a market with 12 sheep to sell. Representatives from three clans — the Black Sheep Clan, the Mud Clan and the Water’s Edge Clan — want to divide each string of sheep. The clans know the sheep will breed, so they are quite interested in getting the right balance of different colored sheep. In particular, the Black Sheep Clan representative wants her strand to be exactly half black sheep, the Mud Clan representative wants exactly half brown sheep, and the Water’s Edge representative wants exactly half white sheep.

In addition, no one wishes to waste rope, so they will only cut the strand in two places to make three different sections.

How can the following strand of sheep be divided to make the three Navajo Clans happy?

A string of twelve sheep. Dave Auckly

One way to divide the sheep would be to give the first four to the Black Sheep Clan, the next six to Water’s Edge, and the final two sheep to the Mud Clan. This way the Black Sheep Clan would get a connected strand of four sheep with exactly two black sheep, Water’s Edge would get a strand of six with exactly three white, and the Mud Clan would get a strand of two with exactly one brown.

Two of the first four sheep are black, three of the next six are white, and one of the last two is brown. Dave Auckly

In fact, with a bit of thought, you can convince yourself that this is the only way to cut this first strand to make each of the clans happy.

Now divide the strands below. Beware! There may be some strands that cannot be divided to make every clan happy. And there may be several acceptable ways to cut and distribute some of the strands.

Can you divide each string of sheep to make the clan representatives happy? Dave Auckly

Bonus: The Single Cut

Let’s say only the Water’s Edge Clan and the Mud Clan were dividing up the sheep. How would you divide each strand with a single cut so that the Mud Clan gets one third brown and Water’s Edge one third white?

For example, the very first strand could be cut once with the first nine sheep (three white) going to Water’s Edge and the last three (one brown) going to the Mud Clan.

That’s it for this week’s challenge. I also asked Dr. Auckly to say a bit about the Navajo Nation Math Circle Project. Here’s his response:

The Navajo Nation Math Circle Project

Numberplay readers have certainly discovered the joy of playing with mathematics. People in the math circle movement believe that one of the best ways to help others learn mathematics is to show them how to have fun playing with mathematical concepts. In an effort to encourage more Native Americans to pursue professions in science, technology, engineering and mathematics, a group of mathematicians formed the Navajo Nation Math Circle project.

This project has several different components. During the academic year, mathematicians visit schools in the Navajo Nation and have students work on puzzles and activities similar to this week’s puzzle.

In addition, the project organizes periodic math festivals where students come to explore math and learn about their culture. The Navajo Circle hosts a two-week summer math camp for middle- and high school students. The circle also has a component for local teachers. The goal of this component is to give the local teachers ideas that they can use to inspire their students to play with math. Some local teachers have already begun after-school math programs for their students.

Thank you, Dr. Auckly! With that we conclude this week’s puzzle. As always, once you’re able to read comments for this post, use Gary Hewitt’s Enhancer to correctly view formulas and graphics. (Click here for an intro.) And send your favorite puzzles to gary.antonick@NYTimes.com.

Solution

Here’s the solution from Steven Lord:

Since the two cuts must have even numbers of sheep on either side, there are only 5 locations where one may cut, and thus 10 cutting schemes: (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), (4,5), where 1 above indicates cutting between the 2nd and 3rd sheep, 2 indicates cuts between the 4th and 5th, etc.. Some simple observations immediately exclude most schemes: e.g. if the first two sheep are of the same color, the first 4 schemes are eliminated, etc.

After eliminating patterns by eye, I get these solutions (where I call halves White, Black, Brown: WBR):

Line 1: cut (2,3) WRB; cut (3,5) WBR
Line 2: no solutions
Line 3: cut (1,4) RWB; cut (1,5) BRW; cut (2,4) RWB; cut (3,4) RWB; cut (4,5) RBW
Line 4: cut (1,5) WRB
Line 5: cut (2,3) WRB or RWB

Bonus: Call the cut schemes now 1,2,3 that leave 3+9, 6+6 and 9+3 sheep. Again observation gives:

Line 1: no solutions
Line 2: no solutions
Line 3: cut (1) RW; cut (2) WR; cut (3) WR
Line 4: cut (1) WR
Line 5: cut (2) RW

I did not find any elegant way to solve this. That is a blanket statement applying to both parts of the problem.

Bravo! Thank you to Dave Auckly and Gordon Hamilton for the puzzle, and thank you to everyone who contributed to the conversation this week: Ravi, Paul, Andrew Ciszewski, Seth Cohen, JosephinBrooklyn, Steven Lord, Stanley Wang and MathPickle.