Fox 354

Roland Sampy has submitted this problem.

It looks right, but not confirmed yet.

Fox 352

We will add comments later...

There are many lazy observations that could help.

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Fox 350

Based on the evidence presented by Bleaug, this is updated as follows:

Thank you.

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Fox 349

This is just one step forward to the general case. Should not be very hard unless the claim is wrong :)

Yes, a geometric solution will be appreciated...

Fox 346

Wooow... This has been quite long time, huh? Life makes people busy (although that is not the exact excuse.)

Is everybody alive? What have been up to? Bleaug? Newzad? Polar Fox ?? ...

The following fox is a simplified version submitted by Roland Sampy. We should be able to ask a general version later. You mean next year??? Who knows hopefully earlier. Here you are...

Oh, by the way the answer was not confirmed yet. So, the claim could be wrong.

A happy new year to everybody.

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Fox 342

Ahmet Arduç submitted this one. Geometric solutions will be appreciated.

http://www.8foxes.com/

Fox 320 - Solutions

This fox has been discussed extensively. See here...

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:

1. Call sub rectangles as TILEs.


2. 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.


3. This will create 0.5 x 0.5 squares (and possibly rectangles around 2 edges -top and/or RHS)


4. 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.


5. Each TILE will have equal areas of black and white (Why?)


6. Therefore the overall rectangle will have equal areas of black and white.


7. 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

3. Induction by Robinson

1. 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)


2. Chose any H-tile, say T(0). (If there is no H-tile, then the result is immediate)


3. If there are H-tiles whose lower border shares a segment with T(0)'s upper border, choose one and call it T(1).


4. 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)


5. 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.


6. Then repeat the same process from T(1). (Continue upward similarly from T(1) to get T(2), and so on...)


7. This will result in a chain of T(0), T(1), T(2), ... , T(m).


8. 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.


9. Remove these tiles and slide the rest together to get a rectangle with fewer H-tiles.


10. Induction applied to this smaller rectangle yields the result for the original rectangle.
    

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Fox 323

Polar Fox: Hey Red, these points seem to have identity issues. Are they black or white?

Red Fox: An obvious illusion! They are white for sure.

Polar Fox: So what I perceive is not the exact reality?

Red Fox: This is just a simple game of colors playing on your retina. It is perfectly explainable and part of the reality.

Polar Fox: Yes, most definitely. But, can this be an artful clue of an even bigger reality that there is truth beyond what you see?

Red Fox: Here we go again!

Polar Fox: A brush mark from a picture? A letter from a book? A note from a symphony?

Red Fox: And who is the artist, juggling with colors to deceive our perception?

Polar Fox: Deceive? I am not so sure. But looks like someone who knows us inside and out. And I think he wants to talk.

Red Fox: Why me? Why me?

Polar Fox: Yep, and you have bigger ears than mine.

Fox 320

Bleaug has a solution for this general case.

http://www.8foxes.com/

Fox 317

Based on our earlier analysis, the sample picture below will never happen.

By "equal-separation" we mean 120-120-120 split.
Otherwise, some may consider 12:00:00 as "equally-separated".

Peace out!

http://www.8foxes.com/

Fox 310

http://www.8foxes.com/

Fox 313

Bleaug submitted the following fox, saying: "a problem which can be expressed in geometric terms and that is elegantly solved using pure algebraic arguments, still giving deep insight."

General case of this problem can be found in literature (with at least 14 proofs :)
But let's try this smaller version here.

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Fox 309 - Solutions

Two solutions to Fox 309...

Bleaug:


Bob Ryden:

Fox 3

We formatted this old one upon César's solution:

http://www.8foxes.com/

Fox 308 - Solution

Bleaug says "a logical follow-up indeed", and solves:

http://www.8foxes.com/

Fox 309

See a similar one (Fox 238). They may be related!

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Fox 308

http://www.8foxes.com/

Fox 306 - Solution

To see the animation below, you will need any of the following browsers:
Chrome, Firefox, Opera, Safari, or IE Explorer 9.0.

Bleaug converts Fox 306 to Fox 302 by "folding" twice. I have seen translation, rotation, similarity, etc., but haven't seen anything like this before. Enjoy and please do respect to human intelligence!

Fox 306

Answer is not confirmed yet.

http://www.8foxes.com/

Fox 305

Yet again,

try to see the goodness,

see the beauty,

surrounding you.

Forget about the numbers, summations, subscripts.

Leave behind the accounts, stocks, papers, statistics.

Just leave yourselves to the arms of an ocean,

full of love and compassion.

Drift away with the blowing wind...

-- Dervish Fox

http://www.8foxes.com/