Key words: Geometry, Unusual geometry, Math, Physics, Chemistry, High school, Geometry Olympiads, Free Geometry, Euclidean Geometry, Calculus, Geometric Construction. Oh yes, going-nowhere discussions, haikus, and poems too.

## Tuesday, January 25, 2011

## Sunday, January 23, 2011

### Fox 325

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## Sunday, January 16, 2011

### Fox 320 - Solutions

**1. Calculus by Bleaug**

He says: "I spotted this problem in a Paul Halmos book "Problems for mathematicians, young and old, 1991", actually in a French translation. He provided the solution below (reformulated by myself) as being proposed by Hugh Montgomery in 1985 in some Math Conference."

**2. Checkerboard Solution by Rochberg and Stein:**

- Call sub rectangles as TILEs.
- Start from the lower left corner of the overall rectangle (let's call this point as the origin)Draw horizontal and vertical lines separated by 1/2=0.5 units starting from the origin.
- This will create 0.5 x 0.5 squares (and possibly rectangles around 2 edges -top and/or RHS)
- Color the first square (that has the origin as one of its corners) as black the next one as white, and so on to generate a checkerboard-look.
- Each TILE will have equal areas of black and white (Why?)
- Therefore the overall rectangle will have equal areas of black and white.
- So overall rectangle has at least one integer side.

**H** means the horizontal side is integer, and **V** means that the vertical side is integer. **a1** is the square at the origin. There could have been a smarter combination, but this simple one illustrates the process clearly. This looks like the closest geometric solution we can get -at this time.

*When there's hardly no day nor hardly no night*

There's things half in shadow and halfway in light

There's things half in shadow and halfway in light

3

**. Induction by Robinson**

- Assume that each H-tile has a width of 1, and each V-tile has a height of 1. Note that any rectangle can be converted this way without distorting the original problem. (This may increase the number of tiles significantly though)
- Chose any H-tile, say T(0). (If there is no H-tile, then the result is immediate)
- If there are H-tiles whose lower border shares a segment with T(0)'s upper border, choose one and call it T(1).
- Otherwise only V-tiles share this border. In this case, we can expand T(0) upward 1 unit. This does not increase the number of H-tiles. Also, the cut V-tiles still have height 1. (They are still V-tiles)
- Continue expanding T(0) until either the top of the rectangle is reached or a choice of an adjacent H-tile T(1) is possible.
- Then repeat the same process from T(1). (Continue upward similarly from T(1) to get T(2), and so on...)
- This will result in a chain of T(0), T(1), T(2), ... , T(m).
- Starting from T(0) again, work downward similarly to obtain a bigger chain:

T(-n), T(1-n), ... , T(0), T(1), ... , T(m-1), T(m) of H-tiles stretching from bottom to top. - Remove these tiles and slide the rest together to get a rectangle with fewer H-tiles.
- Induction applied to this smaller rectangle yields the result for the original rectangle.

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## Sunday, January 9, 2011

### Fox 15

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## Friday, January 7, 2011

### Fox 319 - Solutions

**A solution based on symmetry by Bleaug:**

**Trigonometry and calculus by Six:**

x + y + z = 360 and sinx*siny*sinz

-> f(x,y) = sin(x)*sin(y)*sin(360-x-y)

-> f(x,y) = sin(x)*sin(y)*-sin(x+y)

Now to find the critical points of f(x,y), we just need to find the partial derivative with respect to x and y and solve for 0.

f_x(x,y) =

sin(x)sin(y)(-cos(x+y)-cos(x)sin(y)sin(x+y)

solve for 0.

sin(x)sin(y)(-cos(x+y)-cos(x)sin(y)sin(x+y)=0

-> tan(x) = -tan(x+y)

Since the function is symmetric, we should get the same partial derivative for y.

-> tan(y) = -tan(y+x)

-> tan(x)=tan(y)

-> x = y or they are opposites. However, if they are opposites, the original function just becomes 0. Thus, x = y.

Now substitute in x for y in the original equation and find its critical points.

Eventually, you will get sin(3x)=0

x = 120 degrees

y = 120 degrees

z = 120 degrees

Answer (D)

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## Monday, January 3, 2011

### Fox 324

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## Saturday, January 1, 2011

### Happy new year everybody!

a new beginning,

An unheard adventure - even it makes you late for dinner,

a new beginning.

Cold air that you breath in,

A warm shoulder that you can lean

A promise you've given to yourself - one more time,

a new beginning.

A morning sky painted in the East,

A colorful sunset as it fills the West,

Bread, cheese, water - and the rest,

a new beginning.