Puzzles

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?

I draw the parabola \(y=x^2\) and mark points on the parabola at \(x=17\) and \(x=-6\). I then draw a straight line connecting these two points.

At which value of \(y\) does this line intercept the \(y\)-axis?

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.

What is the area of the largest area triangle that has one side of length 32 and one side of length 19?

The equation \(352x^3-528x^2+90=0\) has three distinct real-valued solutions.

Today's number is the number of integers \(a\) such that the equation \(352x^3-528x^2+a=0\) has three distinct real-valued solutions.

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.