How is 2015 like the Eye of Sauron?

Number Theory Level 2

Happy new year!

2015 = 11111011111 in binary. That is, 2015 is a binary palindrome containing just a single "0" bit -- and thus happens to look a lot like the Eye of Sauron! What's the next number that has this property?

The answer is 8127.

4 solutions

Ossama Ismail
Jan 1, 2015

The next number will be $1111110111111_2 = 1FBF_{16} = 1\times 16^3 + 15 \times 16^2 + 11 \times 16^1 + 15 = 8127$

Dan Wilhelm
Jul 12, 2015

Define the $n$ th palindrome as a $(2n + 1)$ -bit binary number containing all ones except the middle bit. Let the decimal representation of the $n$ th palindrome be given by $P(n)$ ; for example, $P(5) = 2015$ , as given.

The largest number an $n$ -bit string can represent is $(2^n - 1)$ . Hence, a $(2n + 1)$ -bit string of ones represents $(2^{2n+1} - 1)$ . Switching the middle one-bit to a zero results in subtracting $2^n$ , since digits $0$ to $n-1$ are half of the $n$ ones. Hence,

$P(n) = (2^{2n + 1} - 1) - 2^n = 2^n(2^{n+1} - 1) - 1$ .

The next number after $P(5)$ is $P(6) = (2^{13} - 1) - 2^6 = 8127$ .

Bill Bell
Jan 2, 2015

In a Python interpreter:

Brock Brown
Jan 1, 2015

eye = "1111110111111"
total = 0
for i in xrange(len(eye)):
    if eye[i] == '1':
        total += 2**i
print "Answer:", total

How is 2015 like the Eye of Sauron?

The answer is 8127.

4 solutions

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