Puzzles

730

Let \(f(a,b)\) be the number of \(a\)-dimensional sides on a \(b\)-dimensional hypercube. A \((b+1)\)-dimensional hypercube is formed by making two copies of a \(b\)-dimensional hypercube and adding edges between the corresponding vertices on the two copies. Therefore the \((b+1)\)-dimensional hypercube will contain two copies of each \(a\)-dimensional side in the \(b\)-dimensional hypercube; it will also contain a new \(a\)-dimensional side formed by each \((a-1)\)-dimensional side of the \(b\)-dimensional hypercube. Therefore \(f(a,b+1)=2f(a,b)+f(a-1,b)\).

(This can be seen more clearly by looking at the 1-dimensional sides (edges) and 2-dimensional sides (faces) on a 3-dimensional hypercube (cube) and a 2-dimensional hypercube (a square).)

Using the above relation, we find that there are 112 6-dimensional sides on a 8-dimensional hypercube.

153

In general, the expansion of \((x+y+z)^n\) will have \(\frac12n(n-1)\) terms.

285

If I do this for \(x\) and \(y\) from 1 to \(n\) (inclusive) then the result is the sum of the first \(n\) square numbers.

The number can contain no 1 (as all numbers are multiples of 1). It's first and last digits cannot be 2 (as it would then be even and so a multiple of 2).

The lowest three-digit palindrome containing no 0 or 1 and not ending in 2 is 323. This is not a multiple of 2 or 3 and so is the solution.

The product of the numbers in the red boxes is 135.

The squarefree numbered lockers will be locked. (Squarefree number are numbers that are not divisible by any square number apart from 1.) There are 608 squarefree numbers between 1 and 1000.