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25%

18

"You are a slow learner, Winston", said O'Bryan softly. "How can I help it?" he stammered... "How can I help seeing what is in front of my eyes? Two and two are four."

"Sometimes, Winston. Sometimes they are five, but other times they are three. Often they are all of them at once, so you must try harder; It's not easy to become sane."

16

French toast.

Red.

Given square has area of 18 and is half the area of half of the outer square.

Area of half of the outer square 2 *18 = 36 = 1/2 b h. The triangle is a 45/45/90, so b =h, so 36 = 1/2 b^2 or 72 = b^2

So side of outer square, b = SQRT(72)

Diagonal of outer square is the hypotenuse (C) of the 45/45/90 triangle with sides = b = SQRT(72)

b^2 + h^2 = C^2 but b=h, so 2*b^2 = C^2, so C^2 = 2 * (SQRT(72)^2) = 2 * 72 = 144, so then C = SQRT(144) = 12

Side of red square is 1/3 of hypotenuse C = 12/3 = 4

Red square area = 4 x 4 = 16

🤔

Somewhere in here

Blue equals red...

Yellow triangles are 1 each

So confused 🤷‍♂️

18

[video:youtube]https://www.youtube.com/shorts/BUBAHmERkOw[/video]

Adjusted YouTube Shorts video URL to regular YouTube URL.

You’re not the boss of me

Years ago, anthropologist Margaret Mead was asked by a student what she considered to be the first sign of civilization in a culture. The student expected Mead to talk about fishhooks or clay pots or grinding stones.

But no. Mead said that the first sign of civilization in an ancient culture was a femur (thighbone) that had been broken and then healed.

Mead explained that in the animal kingdom, if you break your leg, you die. You cannot run from danger, get to the river for a drink or hunt for food. You are meat for prowling beasts. No animal survives a broken leg long enough for the bone to heal.

A broken femur that has healed is evidence that someone has taken time to stay with the one who fell, has bound up the wound, has carried the person to safety and has tended the person through recovery.

Helping someone else through difficulty is where civilization starts, Mead said.

"We are at our best when we serve others. Be civilized."

18

Let x be the dimension of the lower right inner square.
Let y be half the length of the diagonal of the outer square. y = √2 * x
Let z be the dimension of the red square. z = 2/3 * y = 2/3 * √2 * x
The area of the red square is z2 = 4/9 * 2 * x2 = 8/9 * x2 = 8/9 * 18 = 16.

Gentlemen,

My apologies for resurrecting this somewhat dormant thread but I really enjoy these puzzles. While differences in the answer often leads to a heated debate, I find the differences in method quite fascinating. Consider an additional simple solution using graphical observation.

The outer square is bisected by its diagonal. In the lower right half we have a square and two triangles. The combined area of the two triangles is the same as the square. If the area of the square is A, then the area of the lower right half of the outer square is 2A.

The upper left half contains the red square and three triangles. The combined area of the two larger triangles is the same as the red square. The smaller triangle is the same size and shape as a triangle created by quartering the red square along both diagonals. If the area of the red square is R, then the area of the upper left half of the outer square is 2.25R.

2.25R = 2A ⇒ R = (2/2.25) A = (2/2.25) 18 = 16.

Have a pleasant weekend!