Puzzles

The square numbered lockers will be open at the end of the process. There are 969 non-square numbers between 1 and 1000, so this many lockers will be closed.

121

In general, if you do this with the numbers from \(n+1\) to \(2n\) (inclusive), the result will be \(n^2\) (see this puzzle).

445

By the same method used to solve this puzzle, the answer if 150.

The product of the numbers in the red boxes is 108.

The largest dodecagon will be regular. It's area will be 172.

233

Let \(f(n)\) be the number of ways to write \(n\) as the sum of 1s and 2s. When you write \(n\) as the sum of 1s and 2s, the sum must end in either 1 or 2: if it ends in 1, then there are \(f(n-1)\) ways to write the rest before this 1 as the sum of 1s and 2s; if it ends in 2, there are \(f(n-2)\) ways to write the rest. This means that \(f(n)=f(n-1)+f(n-2)\), or in other words the Fibonacci numbers.

1 Found your far dented horn mixed to make square. (3) — 144 is a square number that is an an anagram of "Found your far dented horn" (one hundred and Forty four).

3 Eno back in Bowie's evening prime. (2) — 17 is a prime that is Eno backwards (onE) followed by a number in Bowie's evening.

4 Prime legs. (2) — 11 is a prime number and called legs in bingo.

5 Palindrome ends cubone, starts ninetales, inside poison ekans. (3) — 191 is a palindrome that is a the end of cubone the start of ninetales and inside poison ekans.

1 Odd confused elven elves hounded deerhound antenna. (5) — 11111 is an odd number that is an anagram of "elven elves hounded deerhound antenna" (eleven thousand one hundred and eleven).

2 Prime try of confused Sven with Beckham's second. (2) — 47 is a prime that is an anagram of "try of" (forty) and Sven with Beckham's second (Seven).

4 Prime even and Ian fed the being. (2) — 19 is a prime number that is the even letters of "and Ian fed the being" (nIneteen).