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2018-02-19 Basic


The animation shows a football player, Josh Imatorbhebhe, jumping high in the air.

He swings his arms in the air and appears to levitate in the air for a brief moment of time.

How does swinging his arms help him accomplish this feat?

Once I went to a fancy restaurant where the ice cubes were clear like a glass of water. When I make them at home, they're opaque and I can't see through them.

The iced drinks I make are never as pretty as the ones at the restaurant. Which of the following approaches might make my ice cubes more clear?

We know that \(\underbrace{4 \times 4 \times 4 \times 4}_{4 \text{ times}} = 4^4.\)

But how many \(4^4\) are there in the left side of the equality \[\underbrace{4^4 \times 4^4 \times \cdots \times 4^4}_{\text{How many times?}} = {\large 4^{\left( 4^4 \right)}} ?\]

There are two identical blue circles and two identical orange circles arranged symmetrically in a larger red circle, as shown. The circles are tangent to each other where they touch.

If the radius of the red circle is \(R,\) then what is the radius of one of the orange circles?

Hint: Try finding a right triangle and applying the Pythagorean theorem.

In the \(MIU\) system, you always start with the string \(MI\) and then form new strings by applying any of following rules any number of times:

1. If your string ends with \(I\), you can add a \(U\) at the end.\(MI\longrightarrow MIU\)
2. You can double the entire string following \(M.\)\(MIU\longrightarrow MIUIU\)
3. Three consecutive \(I\)'s can be replaced with a single \(U.\)\(MIIIU\longrightarrow MUU\)
4. Two consecutive \(U\)'s can be removed from the string.\(MUU\longrightarrow M\)

Which of the following strings can be derived from \(MI\) using these rules?

Note: None of the rules can be used in the opposite way; for example, you aren't allowed to derive \(MUUI\) from \(MI\) by using a reversed version of rule 4.

Source: The \(MIU\) puzzle was introduced by Douglas Hofstadter in his magisterial work Gödel, Escher, Bach.


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