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Algebra Through Puzzles

# Scale and Lever Logic

How much does a single red square weigh?

This Exploration investigates how the logic you intuitively apply when solving puzzles mirrors fundamental algebraic strategies.

To do so, we will teach you new mathematical techniques and push you to recognize and generalize the algebraic techniques that you already intuitively use as you problem-solve.

• A full jar of honey weighs 1000 grams.
• With half of the honey, the total weight is 600 grams.

What is the weight of the empty jar in grams?

What weight will the fourth scale display?

Think carefully! There's a quick and clever shortcut for solving this.

At the end of this Exploration, you'll have a thorough toolbox of algebraic techniques that feel natural to use and empower you to master complex problems and puzzles!

How many green squares should be placed on the right side of the bottom scale in order to make it balance when released?

We’ll also delve into some applications of algebra in physics!

Which way will the balance above tilt if released?

How to Solve Leverage Puzzles:

Archimedes once said:

“Give me a place to stand and with a lever I will move the whole world.”

His claim was in reference to the fact that the amount of force exerted on a lever arm is equal to the weight of the object sitting on it multiplied by the distance of that object from the fulcrum of the lever: $\text{Force} = \text{Weight} \times \text{Distance from Fulcrum}.$ This equation explains why the lever system below is balanced, even though there’s a rock on one side and only a small bird on the other: $$120 \times 1 = 40 \times 3.$$

And, at least in theory, if there were a force pulling the earth and Archimedes both "down" and something fixing the position of the fulcrum in space...

(Yes, Archimedes would also need a space suit.)

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