Puzzles

Today's number is the number \(n\) such that $$\frac{216!\times215!\times214!\times...\times1!}{n!}$$ is a square number.

The equation \(x^2+1512x+414720=0\) has two integer solutions.

Today's number is the number of (positive or negative) integers \(b\) such that \(x^2+bx+414720=0\) has two integer solutions.

Today's number is the number of numbers between 10 and 1,000 that contain no 0, 1, 2 or 3.

Arrange the digits 1-9 in a 3×3 square so: each digit the first row is the number of letters in the (English) name of the previous digit, each digit in the second row is one less than the previous digit, each digit in the third row is a multiple of the previous digit; the second column is an 3-digit even number, and the third column contains one even digit. The number in the first column is today's number.

There is a row of 1000 closed lockers numbered from 1 to 1000 (inclusive). Near the lockers, there is a bucket containing the numbers 1 to 1000 (inclusive) written on scraps of paper.

1000 people then each do the following:

Today's number is the number of lockers that will be closed at the end of this process.

Write down the numbers from 12 to 22 (including 12 and 22). Under each number, write down its largest odd factor*.

Today's number is the sum of all these odd factors.

I make a book by taking 111 sheets of paper, folding them all in half, then stapling them all together through the fold. I then number the pages from 1 to 444.

Today's number is the sum of the two page numbers on the centre spread of my book.

Today's number is the number of 0s that 611! (611×610×...×2×1) ends in.