Puzzles

By rotating the triangles, they can be made to exactly cover the decagon.

The total area of all the triangles is therefore 240.

The first division tells that 12345 is 205 more than a mutliple of today's number, and so today's numwber is a factor of 12140.

The second division tells that 6789 is 112 more than a mutliple of today's number, and so today's number is a factor of 6677.

The only numbers that are factors of both 12240 and 6677 are 1 and 607.

The largest possible isosceles triangles with a perimeter of 50 units is the equilateral triangle with sides of length 50/3. The area of this triangle is 120.28 square-units.

By continuously adjusting the width of the base of the isosceles triangle, triangles with every area between 0 and 120.28 can be created. The animation below shows this.

Triangles with areas of each integer from 1 to 120 can therefore be created. Each area can be made twice: once with a tall and thin triangle, and once with a short and wide triangle. This means that there are 240 different triangles with a perimeter of 50 units and an integer area.

The product of the numbers in the red boxes is 160.

The numbers 10, 20, ..., 1000 end with a 0. There are 100 of these numbers.

The numbers 100, 101, ..., 109, 200, ..., 209, 300, ..., 309, ..., 908, 909, 1000 contain a 0 in their tens column. There are \(10\times9+1=91\) of these numbers.

The number 1000 has a 0 in its hundreds column.

In total, there are 100+91+1=192 zeros.

If \(p_1\), \(p_2\), etc are distinct prime numbers, then the factors of the number \(N=p_1^{a_1}p_2^{a_2}...\) all have the form \(p_1^{b_1}p_2^{b_2}...\), where \(b_1\) is 0 or 1 or ... or \(a_1\) (and similar conditions for \(a_2\), \(a_3\), etc). There are therefore \(a_1+1\) possible values of \(b_1\), \(a_2+1\) possible values of \(b_2), and so on. Therefore, \(N\) has \((a_1+1)(a_2+1)(...)\) factors.

From this we see that the only way to have a number with 37 factors is if it is a prime number to the power of 36, and so \(n=7^{35}\).

\(8n\) is therefore equal to \(2^3\times7^{35}\). This has 4×36=144 factors.

Which other numbers can 37 be replaced with leaving a puzzle that still has a unique solution?

The geometric mean of a number \(N\) is always \(\sqrt{N}\). You can see why this is true by considering the pairs of factors that multiply to make the number: if \(N\) if not square and has \(k\) pairs of factors, then the product of these factors is \(N^k\), so the geometric mean is \((N^k)^{1/(2k)}=N^{1/2}\); if \(N\) is square and has \(k\) pairs of factors plus the square root, then the product of its factors is \(N^k\sqrt{N}=N^{k+1/2}\), so the geometric mean is \((N^{k+1/2})^{1/(2k+1)}=N^{1/2}\). Interestingly, this means that the geometric mean of the factors of a number is an integer only when the number is square.

Therefore the only number whose factors have a geometric mean of 22 is \(22^2\), or 484.

Santa's house could be found by following the directions 3,7,1,1,4,3,3,4,9,3,3,9.