## Ringamatics

Inspired by Michael Huber, who in his new book Mythematics combines math problems with Greek myths, I invented my first logic puzzle. Unlike Huber, I never had any ambition to help Hercules, but I always wanted to assist Frodo.

The day was passing towards sunset when the Company finally caught a long-awaited gleam of water, from which sparkled flickers of sunlight. As they quietly drew nearer, they laid their eyes on the next obstacle — a river that they had to transverse. The Company was footsore and tired and the hobbits were starving. But they couldn’t rest yet. They needed to collect materials with which to construct their raft before it became too dark. By nightfall they managed to build a tiny raft, and eagerly started their supper.

They couldn’t wait until dawn to build more rafts, for they needed to cross the river now. So while they rested, Aragorn smoked his pipe and began to contrive a plan.

Aragorn was in charge and there were eight of them. The four hobbits — Frodo, Sam, Merry and Pippin — were not very useful in battle. However, the four strong fighters — Aragorn, Gimli, Legolas and Boromir, who were sworn to protect the ring-bearer Frodo — were the best in the land.

The small raft they had built would not hold a lot of weight. Aragorn and Boromir were the heaviest. Gimli was short, but together with his armor he weighed as much as either Aragorn or Boromir. Each one of these three heaviest warriors was close to the raft’s maximum capacity, so they had to each be alone on the raft while crossing the river. Among the strong fighters, only Legolas was able to cross the river with a hobbit. The raft could also accommodate two hobbits.

Weight was not Aragorn’s only consideration: the current was dangerously fast. All the strong men could row, but among the hobbits, only Sam was strong enough to row against such a swift current.

Aragorn also worried about the orcs, who were roaming on both sides of the river. He didn’t want to leave any hobbit(s) alone on a riverside, without the safeguard of a strong fighter. Because he was the ring-bearer, Frodo needed extra protection. Aragorn wanted Frodo to be accompanied by at least two strong men. But lately Boromir had become restless when he was around the ring and Aragorn couldn’t count on him to look after Frodo. That is, while on the riverside, Frodo’s protection had to come from two out of the three remaining strong men: Aragorn, Legolas and Gimli.

Can you help Aragorn design a plan to cross the river?

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1. #### Bill:

Cool puzzle! Here’s my solution:

1. Legolas and Sam cross the river first. Frodo is guarded by Aragorn and Gimli.

2. Sam returns alone. Legolas is alone on the other side.

3. Sam rows Merry across. The two hobbits are protected by Legolas.

4. Sam returns alone.

5. Boromir crosses the river to join Legolas and Merry.

6. Legolas returns alone. Boromir now protects Merry.

7. Legolas and Sam cross together.

8. Legolas returns alone. Boromir protects Merry and Sam on the far side.

9. Gimli crosses. Frodo is guarded by Aragorn and Legolas.

10. Sam returns alone.

11. Legolas and Frodo cross. Frodo is now guarded by Legolas and Gimli on the far side. Aragorn protects Pippin and Sam on the bank.

12. Boromir returns.

13. Aragon crosses. Boromir now protects Pippin and Sam on the bank.

14. Legolas returns alone. Frodo is now guarded by Aragorn and Gimli.

15. Legolas and Pippin cross. Sam is still protected by Boromir on the bank.

16. Legolas returns alone.

17. Legolas and Sam cross. Boromir is alone on the bank.

At this point, Boromir may be having some concerns, since he is not being trusted to guard Frodo for some reason, and now finds himself alone with no raft on the wrong side of the river. Nevertheless…

18. Legolas returns!

19. Boromir crosses.

20. Sam returns.

21. Legolas and Sam cross. All are now safely across the river.

2. #### Klemens:

I thought it would be too dangerous to cross the river for two hobbits, so I let them cross river only with Legolas. And Samm can cross it by himself, so there is my solution:
1. Legolas and Sam cross the river
2. Sam comes back
3. Gimli goes on the other side
4. Legolas comes back
5. Legolas and Merry cross
6. Legolas comes back
7. Legolas and Frodo cross
8. Legolas comes back
9. Legolas and Pippin cross
10. Legolas comes back
11. Legolas and Sam cross
12. Sam comes back
13. Aragorn crosses
14. Legolas comes back
15. Legolas and Sam cross
16. Legolas comes back
17. Boromir crosses
18. Sam comes back
19. Legolas and Sam cross. All are at the other side of the river.

Thanks for the puzzle

3. #### Tanya Khovanova:

Sorry, Klemens,

Your solution doesn’t work:for some time Frodo is protected only by Gimli.

4. #### Temmon:

Apologies, I accidentally hit enter too early there. Please delete my previous comment if you can?

So, after spending more time devising a program that doesn’t allow wrong states than I actually spent coming up with a solution, I believe I have a solution. I think it’s really similar to Bill’s except all the hobbits cross as a lump at the end.

1. Sam and Legolas cross
2. Sam comes back (Legolas on bottom, rest on top)
3. Boromir crosses
4. Legolas goes back (Boromir on bottom, rest on top)
5. Sam and Legolas cross
6. Legolas goes back (Boromir and Sam on bottom)
7. Gimli crosses alone (Frodo protected by Aragorn and Legolas, Sam by Boromir and Gimli)
8. Sam goes back
9. Frodo and Legolas cross (Frodo protected by Gimli, Legolas, and Boromir)
10. Boromir goes back (Frodo still protected)
11. Aragorn crosses (Hobbits on top bank protected by Boromir now)
12. Legolas crosses (Hobbits on top still protected, Frodo protected by Gimli and Aragorn)
13. Sam and Legolas cross
14. Legolas goes back
15. Boromir crosses (top bank hobbits protected by Legolas, Frodo is buffered by Gimli and Aragorn from Boromir)
16. Sam goes back
17. Pippin and Sam cross
18. Sam goes back
19. Sam and Merry cross
20. Sam goes back
21. Sam and Legolas cross

And now they can all have a pint! Or kill some orcs, or something.

5. #### Bill:

Temmon,

Your solution does resemble mine, but any correct solution would. Several things need to happen for a successful cross.

1. At some point, Boromir and Sam must be across, while Frodo, Aragorn, Legolas, and Gimli are on the original side with the raft. (I get there in 8 steps, you get there in 6.)

2. Gimli (or Aragorn) crosses. Boromir (or Sam) returns the raft. (My 9 and 10; your 7 and 8.)

3. Legolas and Frodo cross. This is the only way for Frodo to be protected by two out of three non-Boromir strong men at all times. (My 11; your 9.)

4. Sam (or Boromir) returns the raft. (My 12; your 10.)

5. Aragorn (or Gimli) crosses. (My 13; your 11.) This frees up Legolas from guarding Frodo so he can return the raft. It is Legolas who returns the raft because his eventual re-cross can bring a hobbit and not waste a trip.

The hobbits can really cross at any time, as long as they have protection. I sent Merry across early, which put my solution two steps behind yours. I caught up at the end when you had to send Merry across. The fact that we both solved in 21 makes me think the solution is optimal.

6. #### Christ Schlacta:

does none of them have a length of rope? by tying a rope to the raft, any two hobits or 1 man can cross alone without need for a return trip. then gimli can cross alone and the boat pulled back, then legolas and frodo can cross, the boat returned, then the rest can cross in whichever order. I’d recommend boromir be last, only due to his distrustful nature. furthermore, the boat will return faster, and the final passages will be quicker with more people on shore to pull the boat across with the last warrior.