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As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 mins and go!

Question

The operation x&n for all positive integers n greater than 1 is defined in the following manner:

x&n = x raised to the power of x&(n–1).

If x&1 = x, which of the following expressions has the greatest value?

(A) (3&2)&2
(B) 3&(1&3)
(C) (2&3)&2
(D) 2&(2&3)
(E) (2&2)&3

Answer

First, we need to figure out what this strange operation means for a few small integers n. Let’s build upward from 1:

x&1 = x

x&2 = x raised to the power of x&1 (which is just x), so x&2 = x^x = x^x (we’ll use the caret symbol ^ to represent exponentiation, since as we’ll see, we’re going to do it a lot!)

x&3 = x raised to the power of x&2, so x&3 = x^(x^x)

x&4 = x^(x^(x^x))

So the number after the & sign tells you how many x’s are in the exponential expression. Now we can translate the answer choices. As always, do the operation inside the parentheses first.

(A) (3&2)&2

3&2 = 3^3 = 27

27&2 = 27^27 = (3^3)^27 = 3^81

(B) 3&(1&3)

1&3 = 1^(1^1) = 1^1 = 1

3&1 = 3

(C) (2&3)&2

2&3 = 2^(2^2) = 2^4 = 16

16&2 = 16^16 = (2^4)^16 = 2^64

Because both the base and the exponent of this answer choice are smaller, we can tell that choice A is still the winner at this point.

(D) 2&(2&3)

2&3 = 2^(2^2) = 2^4 = 16

2&16 = 2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^2)))))))))))))))

There are sixteen 2’s in this “tower of powers”! This number is incredibly large, far larger than 3^81. Let’s start to collapse the layers to see why.

2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^2)))))))))))))))

= 2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^4))))))))))))))

= 2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^(2^16)))))))))))))

2^16 = 65,536. You aren’t expected to know that, of course, but now imagine 2 raised to that power. This number has thousands of digits.

Now imagine 2 raised to THAT power.

Then 2 raised to THAT power.

And so on, over 10 more times!

This number is the winner by far among the first four answer choices.

(E) (2&2)&3

2&2 = 2^2 = 4

4&3 = 4^(4^4) = 4^(256) = 2^512

While enormous, this number is still far smaller than answer choice (D).

By the way, the operation represented by the & sign in this problem is sometimes called “tetration.” The reason is that just as multiplication is repeated addition, and exponentiation is repeated multiplication, so-called “tetration” is repeated exponentiation. (“Tetra-“ means “four,” and this operation is fourth in line: addition, multiplication, exponentiation, tetration.) Tetration is also called superexponentiation, ultraexponentiation, hyper-4, and power tower.

The correct answer is (D).

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