Puzzles

There are 5 ways to tile a 4×2 rectangle with 2×1 pieces:

How many ways are there to tile a 12×2 rectangle with 2×1 pieces?

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10. Today's number is the product of the numbers in the red boxes.

If \(n\) is 1, 2, 4, or 6 then \((n!-3)/(n-3)\) is an integer. The largest of these numbers is 6.

What is the largest possible value of \(n\) for which \((n!-123)/(n-123)\) is an integer?

190 is the smallest multiple of 10 whose digits add up to 10.

What is the smallest multiple of 15 whose digits add up to 15?

Holly adds up the first six even numbers, then adds on half of the next even number. Her total is 49.

Next, Holly adds up the first \(n\) even numbers then adds on half of the next even number. This time, her total is 465124. What is \(n\)?

Each interior angle of a regular triangle is 60°.

Each interior angle of a different regular polygon is 178°. How many sides does this polygon have?

It's nearly Christmas and something terrible has happened: an evil Christmas-hater has set three drones loose above Santa's stables. As long as the drones are flying around, Santa is unable to take off to deliver presents to children all over the world. You need to help Santa by destroying the drones so that he can deliver presents before Christmas is ruined for everyone.

Each of the three drones was programmed with four integers between 1 and 20 (inclusive): the first two of these are the drone's starting position; the last two give the drone's daily speed. The drones have divided the sky above Santa's stables into a 20 by 20 grid. On 1 December, the drones will be at their starting position. Each day, every drone will add the first number in their daily speed to their horizontal position, and the second number to their vertical position. If the drone's position in either direction becomes greater than 20, the drone will subtract 20 from their position in that direction. Midnight in Santa's special Advent timezone is at 5am GMT, and so the day will change and the drones will all move at 5am GMT. For example, if a drone's starting position was (1, 12) and its movement was (5, 7), then:

You need to calculate each drone's starting position and daily speed, then work out where the drone currently is so you can shoot it down.

You can attempt to shoot down the drones here.

The expression \((3x-1)^2\) can be expanded to give \(9x^2-6x+1\). The sum of the coefficients in this expansion is \(9-6+1=4\).

What is the sum of the coefficients in the expansion of \((3x-1)^7\)?