Does 64 = 65?

This post was updated on Feb. 26, 2007 to fix the links.

Watch this animation [found via MeFi, link opens in a new window]. Intriguing, no? After watching it the first time, I thought it was pretty neat, but knew it wasn’t possible. So I did the math, and even showed my work.

It’s an optical illusion, powered by the outlines on the sections they move. First, they make two slices and rearrange things a bit. The blue slice creates two triangles, and the red slice creates two trapezoids.

Rearrange as desired, and without any outlines to cover things up, this is what you get:

See that white gap? It’s a parallelogram with an area of 1, which is why 64 seemingly equals 65. Warning: geometry and trig follow.

Start off noting that the two blue triangles are identical, as are the two red trapezoids.

For the blue triangle: looking at the smallest angle, you know the adjacent side is 8, and the opposite side is 3.

32 + 82 = x2, so the hypotenuse is 8.544
tan-1(3/8) = 20.556°

For the red trapezoid: look at the slice that happened. Basically, it took a triangle (green) out of a square to make the red trapezoid.

On that triangular piece, looking at the smallest angle, you know the adjacent side is 5, and the opposite side is 2.

22 + 52 = y2, so the hypotenuse is 5.385
tan-1(2/5) = 21.80°

Given that it’s the slice that was removed from the 5 x 5 square to result in the red trapezoid, you know the acute angle of the trapezoid and the length of the slanty side.

90° – 21.80° = 68.20°
Length of slanty side = removed section hypotenuse = 5.385

Now you know the acute angle of the parallelogram in the middle.

90° – 68.2° – 20.556° = 1.244°

So now you know everything you need to find the area of the parallelogram: opposite sides blue are 8.544, opposite sides red are 5.385, and the acute angles are 1.244°.

Area of a parallelogram is base times height, so you need to draw a right triangle inside. The following diagram is totally skewed for clarity, but the values are accurate.

You know the angle is 1.244°, and the hypotenuse of your newly drawn right triangle is 5.385. So you know the height of the parallelogram.

sin(1.244) x 5.385 = 0.1169

Knowing that the base is 8.544, you now know the area of the parallelogram.

8.544 x 0.1169 = 0.9988

… which is close enough to 1 to explain the illusion, because I rounded some things and didn’t follow the rule of significant digits.

QED, bitchez. :) Not that you care.


15 thoughts on “Does 64 = 65?

  1. How many people know the puzzle involving the spending of $100.00 for 100 birds with the condition that you buy at least one of each, as follows, chicks at 5 cents, hens for $1.00 and roosters for $5.00 each. You have to spend $100.00 for 100 birds. How many of each do you buy? This was probably published somewhere but I have no idea when or where.

    Is anyone familiar with the pentominoes puzzles, or is it spelled pentaminoes? This is one of my favorites.

    Here is one of my poems:

    WARNING: This poem is not written or

    approved by George Bush.

    George Bush (9/3/04)

    I’m George Bush and you should vote for me,

    so I can export your job across the sea.

    The rich deserve to have even more.

    All the poor can move offshore.

    Send contributions and repeat my pitch.

    I take from the poor and give to the rich.

    I’m really a puppet on a string,

    too dumb to understand anything.

    Your religion should be the same as mine.

    Always adhere to the party line.

    Help pull my campaign wagon. Grab the hitch.

    I take from the poor and give to the rich.

    Social security and medicare have to go,

    so big corporation’s profits can grow.

    All sstudents will be left far behind,

    when I cut funding of every kind.

    I want to control every master switch.

    I take from the poor and give to the rich.

    Never remember when I make a mistake.

    The rich deserve whatever they can take.

    Let’s have logging and dams in every park.

    The unemployed can freeze in the dark.

    Turn in your neighbor and become a snitch.

    I take from the poor and give to the rich.

    (copyright 2004 by Norman Riger)

  2. 80 chicks ($4)

    19 roosters ($95)

    1 hen ($1)

    This problem has 3 variables and only 2 equations. But, you can get rid of one variable and get something like:

    roosters = chicks*95/400

    but from there you have to plug in numbers. It is easy because the only possibilties are 20,40,60 and 80 (otherwise you don’t get an even dollar amount). 80 is the only one that gives a whole number for the # of roosters.

  3. As for this one:

    It’s virtually the same as the first problem posed. The hypotenuse of the larger triangle is not a straight line, so it isn’t really a triangle at all. The green part has a slope of 0.4. The red part has a slope of 0.375.

    In the top picture, it bows in slightly. In the lower picture it bows out.

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