El Cuadrado Perdido

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W EDNESDAY, JULY 22 , 2 009

Solution to the "Missing Square Problem"Solution of the Missing Square ProblemJa son D.

Ha ve y ou ev er seen this "puzzle" floating aro und before?

This problem appears to make no sense what so ev ertwo triangles of apparently equal dimensions, consisting of smaller segments, appearto cov er less area when rearranged differently...where did the missing

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Solution to th e"Missing Squar eProblem"

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About MeJA Y I'm a friendly , lov able,goofy geek w ho likes rock m usic, com puter science,science fiction/fanta sy ,good conv ersation and tohelp people whenev er Ican .

V ie w m y com pl et e p r ofi le

blo ck go?

Go ahead and try it for yo urself (I used cut outs from graph paper as astarting point). Then come bac k and see if yo ur solution agrees withmine!

The solution:

First, lets examine the larger rec tangles. Each co ver s and area of (13 *5)/2, or 32.5 square units.

The areas of the individual segements:

Blue: (2*5)/2 = 5 square unitsGreen: 8 square units Y ellow: 7 square unitsRed: (3*8)/2 = 12 square units.

The sum? 32 square units. *GASP* neither triangle co rresponds to thesum area o f the segments.

In the top triangle, the area subtract the area of the yellow and greensegments gives us 17 .5 square units.

In the botto m triangle, this identical calculation y ields 16.5 squareunits.

The different of course being *drum ro ll* one square unit. But WAIT!

There's mo re.

The ratio of the full triangle is 1 3:5, the blue 5:2 and the red 8:3. Theseare NOT equivalent ratios.

We can look to o ur good friend Py thagoras:

Given the hypo tenuse of each triangle is the square root o f the sum of the other two sides squared, the hypotenuse o f each triangle is:
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Full Triangle: 194

Red: 7 3Blue: 29

If this triangle is all that it seems, the hy potenuse o f the red and blue tr iangle would equal that of the full tr iangle.

73 = 8.544003745317 531167 8716483262397+ 29 = 5.385164807 1345040312507 104915403= 13.92916855245203519912235881778

194 = 13.928388277 1841193384677 38928513

So close! But the facto r is in fact o ff by .00078027526791586065461988926674228

Lets look closely at the slopes.Full Triangle : 5/13 = 0.384615384615 384615384615 38461538Red Triangle : 3/8 = 0.37 5Blue Triangle: 2/5 = 0.4

If this were what it loo ked like, then all the slopes wo uld be equal.Howev er, The blue triangle is steeper then the red, which is not assteep as the full triangle. This means that the hy potenuse formed by the two smaller triangles is not straight, and thus slopes in on one

triangle and slopes out on the o ther.

The hypotenuses have different slopes, as the angles are different.

For n >= 5, this discrepancy is basically unnoticeable. But for n=4,n=3, yo u can see it quite clearly .
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From a different perspectiv e:
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Still not conv inced?

Pretty nifty . But wait just a darn to otin' minutelook at thosenumbers 1, 1 , 2, 3, 5, 8, 13

Look familiar?

So if youre still wondering where the missing block went, Fibonacc iate it.
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The End.Posted by Jay at 1:38 PM Labels: cool , illusion , internet , math , puzzle , solutions

3 comments:

SmokeStacked Oct 4, 2011 09:27 AMThanks, you did a fan-freakin tastic job of explaining it.

Reply

quoroy Mar 22, 201 2 08:56 AM

very interesting! i had been searching for the answer to thisproblem for a long time now thanks!

Reply

elikz Mar 24, 2012 1 2:46 PM

fibonacci ate it xD ... nice conclussion to the problem... niceexplanation

Reply
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