Puzzles

There are 6 terms in the expansion of \((x+y+z)^2\):

Today's number is number of terms in the expansion of \((x+y+z)^{16}\).

For \(x\) and \(y\) between 1 and 9 (including 1 and 9), I write a number at the co-ordinate \((x,y)\): if \(x\lt y\), I write \(x\); if not, I write \(y\).

Today's number is the sum of the 81 numbers that I have written.

Arrange the digits 1-9 in a 3×3 square so that the first row makes a triangle number, the second row's digits are all even, the third row's digits are all odd; the first column makes a square number, and the second column makes a cube number. The number in the third column is today's number.

Today's number is smallest three digit palindrome whose digits are all non-zero, and that is not divisible by any 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 product of the numbers in the red boxes.

There is a row of 1000 lockers numbered from 1 to 1000. Locker 1 is closed and locked and the rest are open.

A queue of people each do the following (until all the lockers are closed):

Today's number is the number of lockers that are locked at the end of the process.

There are 2600 different ways to pick three vertices of a regular 26-sided shape. Sometimes the three vertices you pick form a right angled triangle.

Today's number is the number of different ways to pick three vertices of a regular 26-sided shape so that the three vertices make a right angled triangle.

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