Advent calendar 2021

Reindeer 1's favourite food is food 1.

Reindeer 2's favourite food is food 4.

Reindeer 3's favourite food is food 3.

Reindeer 4's favourite food is food 1.

Reindeer 5's favourite food is food 9.

Reindeer 6's favourite food is food 6.

Reindeer 7's favourite food is food 8.

Reindeer 8's favourite food is food 3.

Reindeer 9's favourite food is food 5.

20 can be written as the product of one-digit numbers greater than 1 in the following ways:

Our 12-digit numbers must contain one of these plus lots of 1s.

In the first case, there are 12×11=132 different possible positions for the 4 and the 5.

In the second case, there are 12×11×10×½=660 different possible positions for the 2s and the 5.

In total, this makes 132+660=792 numbers.

The line will meet the \(y\)-axis at y=102.

The two tokens must be in two non-adjacent columns. There are ½×21×20 ways of picking two different columns. 20 of these ways will give two adjacent columns, so there are ½×21×20–20=190 ways to pick the columns.

Once the columns are picked there are four choices for the rows to place the tokens in (up and up, up and down, down and up, down and down). 4×190=760.

Today's number is 148.

The triangle will be largest when the two sides meet at a right angle. The area of this triangle is ½×32×19=304.

The function \(f(x)=352x^3-528x^2+a\) with \(a=90\) looks like this (click to enlarge):

Adjusting the value of \(a\) will move the curve up and down. The equation \(f(x)=0\) will have three distinct solutions as long as the local minimum is below the \(x\)-axis and the local maximum is above the \(x\)-axis.

The minimum and maximum are at \(x=1\) and \(x=0\). The difference between \(f(1)\) and \(f(0)\) is 176. This means that there are 175 possible integer values of \(a\) (as the endpoints are not included).

The product of the numbers in the red boxes is 432.

252 can be written as the product of one-digit numbers in the following ways:

Therfore, the smallest number whose digital product is 252 is 479.

Today's number is 222.

The mean of the numbers in the \(n\)th row is \(n^2\), so the mean of the number in the 19th row is 361.

The route must contain three upward movements. As the same line cannot be travelled along twice, there is only one possibility for the rest of the route. There are 8 choices for the position of each upward movement, so there are 8×8×8=512 possible routes.

The two triangles are similar, and the sides of the blue triangle are twice the length of the red triangle. Therefore the area of the blue triangle is 4 times the area of the red triangle. 226×4=904.

The path from A to B will consist of 10 moves. 3 of these moves must be moves up. Therefore there are "10 choose 3" possible routes. 10 choose 3 is 120.

The rightmost number is the \(n\)th row is \(n^2\). 750 is between \(27^2\) and \(28^2\) so is in the 28th row.

The numbers in the \(n\)th row are \(2n\) less than the numbers below them, and so the number above 750 is 2×27 less than 750. 750–2×27=696.

The largest number you can make using the digits in the red boxes is 654.

If Eve had written down an even number of numbers, then the mean of her numbers would be halfway between the two middle integers, so would be an integer and a half.

If Eve had written down two numbers, then the mean of those two numbers would be 422. This is not an integer and a half.

If Eve had written down three numbers, then the mean of those three numbers would be 281.3333. This is not an integer.

If Eve had written down four numbers, then the mean of those two numbers would be 211. This is not an integer and a half.

If Eve had written down five numbers, then the mean of those three numbers would be 168.8. This is not an integer.

If Eve had written down six numbers, then the mean of those two numbers would be 140.6666 This is an not integer and a half.

If Eve had written down seven numbers, then the mean of those two numbers would be 120.5714. This is an not integer.

If Eve had written down eight numbers, then the mean of those two numbers would be 105.5. This is an integer and a half, so this is possible. The eight consecutive integers with a mean of 105.5 are 105 and 106, 104 and 107, 103 and 108, and 102 and 109.

To be certain that this is the only answer, we should rule out Eve having written more than eight numbers.

The sum of the numbers from 1 to 41 is 861. This is greater than 844, so Eve must have written fewer than 41 numbers.

If Eve had written down an odd number of numbers, then the mean of her numbers would be an integer. This means that the number of numbers would be a factor of the sum of all the numbers. 844's prime factorisation is 2×2×211, and so its only odd factor is 211. This is greater than 41, so Eve didn't write down this many numbers.

Eve therefore must have written down an even number of numbers.

If Eve wrote down an even number of numbers, then the mean of her numbers would be an integer plus a half. This implies that half the number of numbers would be a factor of the sum, and the number of numbers cannot be a factor of the sum. This is only possible if there are 8 numbers.

836=2×2×11×19, and so for the product to be correct the integers could be:

The only one of these whose sum is 51 are 2, 11, and 38. Therefore the product of the largest and second-largest integers is 418

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.