Geometry Question from Dinner

[calcbandit]

So I'm eating dinner at a restauraunt and looking at the floor. The floor is tiled. Say vertical and horizontal lines with 1 foot spacing between each line.

Then I saw a chair, standard 4 legs. The spacing between the legs is 1.5 ft in each direction.

Immediately I started looking and trying to twist the chair on the ground in mind to see if I could get all 4 legs of the chair to be on one of the lines for the tiled floor.

1) Is it possible? What angle and X,Y coordinate should the first chair leg be at given that (0,0) is the start of the first horizontal and vertical line (if there is 1 answer there are infinite answers unless you constrain the # of tiles. just give one answer)

2) If no answer, why not?

3) Are there answers for every tile and chair spacing? Obviously if the tiles are 1 foot apart and the chair is 2 feet apart then it is going to work. How can you tell which values will work and which won't.


K thx don't make fun of me I couldn't help wondering.

[PokrLikeItsProse]

I think it should be solvable, although I'm not going to solve for the answer.

Basically, you have a square with sides of s=1.5. You need to place it in a square with sides of n, where n is a multiple of the sides of the floor squares, so that the corners of the smaller square touch the sides of larger square.

This will divide the larger square into the smaller square and four right triangles with sides of x, y, and s.

So you have equations:
x-squared + y-squared = s-squared
x + y = n

Given values for s and n, I think that should be a solvable equation.

[Alex-db]

I believe that if the centre of the chair is above the centre of a grid square, it is always possible to rotate the chair until the 4 legs simultaniously meet the grid lines.

Similarly you can always place 4 legs of a square chair on just 2 lines which are at right-angles, which will always be a part of the grid. Does this meet your rules?

[jay_shark]

Suppose the floor tiling is spaced a units apart . The spacing between the legs of a chair is x units . For simplicity , put one of the legs on the origin (0,0) of the xy plane and pivot around this point . Fix a second leg on a horizontal line and note that this may occur more than once ;to be precise , this will happen at most [x/a] times .

Once we fix a second leg on one of the horizontal lines, say the first line, this new coordinate becomes (a,b) for some point b . The third and fourth coordinates are now easily determined using vectors which is (b,-a) and (b+a,-a+b) . So again , if this were possible for this specific configuration , then b must be a multiple of a .

Say b=ca for some integer c . Therefore we wish to solve for the equation a^2 + (ca)^2 = x^2 . If a and x are given in the question , then we can solve for the equation which becomes a variable in c and MUST be an integer .

Going back to our previous example , we know that it isn't possible since 1^2 + (c*1)^2 = 1.5^2 is a contradiction for some integer c .

[danny]

Put center of chair over intersection of lines. Rotate 45 degrees.

[BruceZ]

[ QUOTE ]
So I'm eating dinner at a restauraunt and looking at the floor. The floor is tiled. Say vertical and horizontal lines with 1 foot spacing between each line.

Then I saw a chair, standard 4 legs. The spacing between the legs is 1.5 ft in each direction.

Immediately I started looking and trying to twist the chair on the ground in mind to see if I could get all 4 legs of the chair to be on one of the lines for the tiled floor.

1) Is it possible? What angle and X,Y coordinate should the first chair leg be at given that (0,0) is the start of the first horizontal and vertical line (if there is 1 answer there are infinite answers unless you constrain the # of tiles. just give one answer)

[/ QUOTE ]

It is possible by placing the chair legs at these coordinates:

(1/sqrt(8), 0)

(1, 1+1/sqrt(8))

(-1/sqrt(8), 2)

(-1, 1-1/sqrt(8))

[jay_shark]

If you're allowed to use 2 vertical and 2 horizontal lines then it is possible like BruceZ showed .

However , it isn't possible if you restrict the question to either all 4 horizontal lines or 4 vertical lines which my answer shows . At least that is how I interpreted the question .

[Silent A]

[ QUOTE ]
Put center of chair over intersection of lines. Rotate 45 degrees.

[/ QUOTE ]

Yeah, this was my solution. All sized tiles and chairs work for this.