Puzzles

It's nearly Christmas and something terrible has happened: you've just landed in a town in the Arctic circle with a massive bag of letters for Santa, but you've lost to instructions for how to get to Santa's house near the north pole. You need to work out where he lives and deliver the letters to him before Christmas is ruined for everyone.

Due to magnetic compasses being hard to use near the north pole, you brought with you a special Advent compass. This compass has nine numbered directions. Santa has given the residents of the town clues about a sequence of directions that will lead to his house; but in order to keep his location secret from present thieves, he gave each resident two clues: one clue is true, and one clue is false.

Here are the clues:

You can view the map here.

There are six ways to put two tokens in a 3 by 3 grid so that the diagonal from the top left to the bottom right is a line of symmetry:

Today's number is the number of ways of placing two tokens in a 29 by 29 grid so that the diagonal from the top left to the bottom right is a line of symmetry.

198 is the smallest number that is equal to 11 times the sum of its digits.

Today's number is the smallest number that is equal to 48 times the sum of its digits.

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 largest number you can make with the digits in the red boxes.

There are 3 ways to order the numbers 1 to 3 so that no number immediately follows the number one less that itself:

Today's number is the number of ways to order the numbers 1 to 6 so that no number immediately follows the number one less that itself.

18 can be written as the sum of 3 consecutive (strictly) positive integers: 5 + 6 + 7.

18 can also be written as the sum of 4 consecutive (strictly) positive integers: 3 + 4 + 5 + 6.

18 is in fact the smallest number that can be written as the sum of both 3 and 4 consecutive (strictly) positive integers.

Today's number is the smallest number that can be written as the sum of both 12 and 13 consecutive (strictly) positive integers.

The diagram to the right shows a triangle. Two of the sides of the triangle have been split into three pieces, with lines drawn from the opposite vertex. In total, the diagram now contains 27 triangles of any size.

Another triangle has two of its sides split into eight pieces, with lines drawn from the opposite vertex. How many triangles (of any size) would this create?

The expansion of \((x+y+z)^3\) is

This has 10 terms.

Today's number is the number of terms in the expansion of \((x+y+z)^{26}\).