Problem 935

[serhkov]

Problem 935. Problem text has F(6)=4.
But, for all values of b<=0.2 the small square has 4 steps to return to its initial position. That's why F(6)>>4. ?

[DJohn]

for all values of b<=0.2 the small square has 4 steps to return to its initial position.

Let's say b = 0.1. One step will take it from its initial position to a new position 0.1 along the bottom of the large square. After four steps it will still be somewhere on the bottom edge, and it will take many more to return to its original position (it has to go all the way along all four sides of the larger square).

Four steps will return it to its original orientation, but that's not what the problem is asking about.

[serhkov]

I can't agree, because the square with b=1/2 finishes its four steps at third side(this is example from the problem 935)

[DJohn]

1/2 > 0.2, so that's not what I'm talking about, and it's not what your initial post was about.

If b = 1/2, then it does indeed take four steps to return to its original position. I don't know what you mean by "at its third side" - the problem is talking about the position of the small square within the large square, and doesn't distinguish between the sides of the small square.

For b = 1/2: it starts in the bottom left corner. The first step takes it to the bottom right corner (and since the sides of the small square are exactly half the length of the sides of the large square, it is fitting exactly in that corner, with two of its sides in line with bottom and right sides of the large square). The second step takes it to the top right, the third to the top left, then the fourth to the bottom left. At each step it rotates 90 degrees (it always lands neatly in the corners, so there are no odd angles like in the b = 5/13 example).

If b < 0.2, then the first few steps will have the small square rolling along the bottom sides of the large square. It can't possibly return to its original position in the bottom left corner after only four steps - it won't even get to the bottom right corner. If you think that any small square with b < 0.2 will return to its initial position in four steps, it would help if you describe how you think it moves for a particular value - I suggest b = 0.1. You're definitely misunderstanding something, but I can't identify what.

[serhkov]

I don't agree with your words at the first post: "(it has to go all the way along all four sides of the larger square)". The square with b=1/2 finished its way (four steps - four positive positions) at the up side of larger square and does not go along all sides.

[mdean]

Respectfully, just because you disagree with him, it doesn't mean he's not right. b=$\frac12$ happens as he says it does. If the smaller square's initial position is sharing the lower left corner of the larger square, after 4 steps, it's back to sharing the lower left corner.

[serhkov]

You are right. Thanks for the communication.