Wednesday, July 22, 2009

Solution to the "Missing Square Problem"

Solution of the Missing Square Problem
Jason D.

Have you ever seen this "puzzle" floating around before?

This problem appears to make no sense what so ever…two triangles of apparently equal dimensions, consisting of smaller segments, appear to cover less area when rearranged differently...where did the missing block go?

Go ahead and try it for yourself (I used cut outs from graph paper as a starting point). Then come back and see if your solution agrees with mine!

The solution:

First, lets examine the larger rectangles. Each covers and area of (13 * 5)/2, or 32.5 square units.

The areas of the individual segements:

Blue: (2*5)/2 = 5 square units
Green: 8 square units
Yellow: 7 square units
Red: (3*8)/2 = 12 square units.

The sum? 32 square units. *GASP* neither triangle corresponds to the sum area of the segments.

In the top triangle, the area subtract the area of the yellow and green segments gives us 17.5 square units.

In the bottom triangle, this identical calculation yields 16.5 square units.

The different of course being *drum roll* one square unit. But WAIT! There's more.

The ratio of the full triangle is 13:5, the blue 5:2 and the red 8:3. These are NOT equivalent ratios.

We can look to our good friend Pythagoras:

Given the hypotenuse of each triangle is the square root of the sum of the other two sides squared, the hypotenuse of each triangle is:

Full Triangle: √194

Red: √73
Blue: √29

If this “triangle” is all that it seems, the hypotenuse of the red and blue triangle would equal that of the full triangle.

√73 = 8.5440037453175311678716483262397
+ √29 = 5.3851648071345040312507104915403
= 13.92916855245203519912235881778

√194 = 13.928388277184119338467738928513

So close! But the value is actually off by .00078027526791586065461988926674228

Lets look closely at the slopes.
Full Triangle : 5/13 = 0.38461538461538461538461538461538
Red Triangle : 3/8 = 0.375
Blue Triangle: 2/5 = 0.4

If this were what it looked like, then all the slopes would be equal. However, The blue triangle is steeper then the red, which is not as steep as the full triangle. This means that the hypotenuse formed by the two smaller triangles is not straight, and thus “slopes in” on one triangle and “slopes out” on the other.

The hypotenuse’s have different slopes, as the angles are different.

For n >= 5, this discrepancy is basically unnoticeable. But for n=4, n=3, you can see it quite clearly.

From a different perspective:

Still not convinced?

Pretty nifty. But wait just a darn tootin' minute…look at those numbers… 1, 1, 2, 3, 5, 8, 13

Look familiar?

So if you’re still wondering where the missing block went, Fibonacci ate it.

The End.

Saturday, July 18, 2009

Funny Out-Of-Office E-Mail Auto-Replys

Out of office replies can either be really useful (for you) and really annoying (for the people getting the reply).

To keep them from getting too angry (or not angry enough), here are some good responses to provide to people looking to contact you with some humor before you get back (hopefully).


Thursday, July 9, 2009

Office 2010: The Movie

Hi all!

Saw this today on Beyond Binary, and just had to share it. Probably one of the best commercials I've seen a long time, and not just for computer products. I had a grin on my face the whole time, who know Office could be so exciting? Well, there was that flight simulator back in the day...

Anyway, check out the full video at the link above or directly on YouTube. And don't forget to check out the official site.