# Brain Teaser #87

Brain Thumper #2

Three reasonably thin ropes of more than sufficient length are specially made so that they connect seamlessly at the ends. One is red, one white, and one blue. They are floated in a large kiddie pool which is then drained for the following game.

A person picks a location in the pool to start and puts on a pair of clogs. She must walk forward, heel-to-toe, only touching the rope by stepping directly on it so as to cross over. Each time she does this it is called a clog.

Another player at the side of the pool has three standard dice, red, white, and blue. He may toss any die in his hand but cannot pick it up again. When he rolls, the player in the pool must make the number of clogs shown on the die without crossing over rope of a different color. If she cannot, she loses.

For instance, if a 2 is rolled with the white die, she must walk, placing one foot immediately in front of the other, such that the front and back ends of a clog span the white rope on exactly two of her steps. She may, for instance, cross a stretch of white rope, take a few steps to turn around, and cross again. Since she may not cross red or blue rope, in this case she has returned to where she started. If she were to find herself surrounded by only red and blue rope, she would lose.

Additionally, she must reach the side of the pool at the end of the third throw to win. If she does not, the players switch roles, resetting the game.

Suppose you get to go first and your brutally competitive friend rolls. Assuming you're not a klutz, is there a configuration of rope that guarantees a win?

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Are the ropes connected so as to form a closed loop?

Are the red, white & blue segments equal in length?

gnome said:
Are the red, white & blue segments equal in length?
I'm not sure what you mean by segments, as each rope forms a closed loop, to answer your first question. This was poorly stated on my part. If you wish, they may be of equal length. I don't think it changes the problem.

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Njorl
I would say no, but that is dependent on a few assumptions.

The optimal configuration would be like a Ballantine Beer logo. Interlace the three circles so there are 3 single color regions, 3 2-color overlaps, and one 3-color overlap. Even this fails.

I am assuming that a player can not be "on a line". What I mean is, the walking person is always considered to be inside a circle, crossing a circle, or outside the circle. They can't walk around the edge of a circle like a tightrope.

Just consider the final throw. You are either inside the relevent circle or outside. After all of your crossings, you must be outside the circle. If you start inside, you need an odd number. If you are outside, you need an even number. No guaranteed win is possible.

This would be possible if the rings extended beyond the edge of the pool, but I assume the statement that they were floated precludes this.

Winning would be possible to guarantee if all three dice were rolled before you picked your starting point.

Njorl

I'm not sure what you mean by segments
I thought you meant the ropes were connected to each other, end-to-end.

Of course, with three separate loops of rope, a win is guaranteed if each of the ropes falls in a figure-8, none of the central intersections of the "eights" is located inside a loop of another color, and a sufficient portion of each rope is located outside of both of the other ropes to provide space in which to walk without stepping on another rope. Or, to make it simpler, no rope is enclosed by another rope.

The player in the pool starts outside the ropes, and if an odd number is thrown, takes one of the required clogs (or the only one, if a 1 is thrown) across the central intersection of the appropriate rope.

The rest is obvious.

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Njorl said:
I am assuming that a player cannot be "on a line".
That is correct. The player can only touch the rope so as to cross over.

Njorl said:
I assume the statement that [the ropes] were floated precludes [extending] beyond the edge of the pool.
Correct. The player must be outside all loops to win, or the other gets to try.

Njorl said:
...if all three dice were rolled before you picked your starting point
That is not the case. The starting point is selected first, and the dice are thrown one at a time.

Njorl said:
Just consider the final throw. You are either inside the relevent circle or outside. After all of your crossings, you must be outside the circle.

If you start inside, you need an odd number. If you are outside, you need an even number.
Unfortunately you're making an assumption about the configuration.

gnome said:
A win is guaranteed if each of the loops falls in a figure-8.... The player in the pool starts outside the ropes, and if an odd number is thrown, takes one of the required clogs...across the central intersection of the appropriate rope.
gnome with the point!

I'm so ashamed. My original solution was ten times more complicated.

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NateTG
Homework Helper
Clearly if the pool is small, for example less than 1 clog in diameter, then it is impossible.
For larger pools, it depends on what you mean by arrangement, and how the rules work. If the player is allowed to cross ropes between segments, or if the arragement allows branching, then there are patterns where any combination of results work.
Complications that occur, and are not addressed in your description, are, for example a question whether:
Each length of 1 clog travesed must be straight, and whether it's legal to double back accross a segment of a length of 1.

I don't know if it makes sense to discuss this after the puzzle has been solved, but I'll go ahead and address your questions anyways.

NateTG said:
Clearly if the pool is small, for example less than 1 clog in diameter, then it is impossible.
The problem clearly stated that it is a large kiddie pool.

NateTG said:
if the player is allowed to cross ropes between segments
Are you asking if the player is allowed to change her position between rolls? It wasn't specifically addressed, but the common sense answer is correct. I'm sorry the description isn't more formal. You must understand that I was criticized earlier for not expressing myself in plain English.

Are you asking about other rope surrounding the player? She may not touch rope of a color different from the die, so walking heel-to-toe it is impossible to cross this rope. However, she may cross the indicated rope wherever it is accessible to her.

NateTG said:
if the arragement allows branching
Each of the ropes is a single continuous loop as clarified by gnome's question.

NateTG said:
whether...each length of 1 clog travesed must be straight
The problem suggested in other ways that, at the point it is crossed, the rope must be sufficiently straight. For instance, the front and back of the clog must span over the rope. You are aware of what clogs look like, aren't you? Also, by being floated in the pool, we can eliminate the possibility of sharp angles in the rope, although I'm not quite sure what to make of it if the rope were to get tangled.

NateTG said:
whether it's legal to double back accross a segment of a length of 1
This is legal provided the player's motion is always forward, but I don't see how it changes the problem. As stated, the rope is of more than sufficient length, so you shouldn't be concerned about limiting cases.

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