Puzzles

Write the numbers 1 to 81 in a grid like this:

Pick 9 numbers so that you have exactly one number in each row and one number in each column, and find their sum. What is the largest value you can get?

The digital product of a number is computed by multiplying together all of its digits. For example, the digital product of 1522 is 20.

How many 12-digit numbers are there whose digital product is 20?

There are 12 ways of placing 2 tokens on a 2×4 grid so that no two tokens are next to each other horizontally, vertically or diagonally:

Today's number is the number of ways of placing 2 tokens on a 2×21 grid so that no two tokens are next to each other horizontally, vertically or diagonally.

Arrange the digits 1–9 (using each digit exactly once) so that the three digit number in: the middle row is a prime number; the bottom row is a square number; the left column is a cube number; the middle column is an odd number; the right column is a multiple of 11. The 3-digit number in the first row is today's number.

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.

The digital product of a number is computed by multiplying together all of its digits. For example, the digital product of 6273 is 252.

Today's number is the smallest number whose digital product is 252.

The odd numbers are written in a pyramid.

What is the mean of the numbers in the 19th row?

The integers are written in a triangle as shown below:

Today's number appears directly above the number 750 in the triangle of integers.