Puzzles

Today's number is a multiple of three. The average (mean) of all the answers that are multiples of three is a multiple of 193.

Earlier this year, I wrote a blog post about different ways to prove Pythagoras' theorem. Today's puzzle uses Pythagoras' theorem.

Start with a line of length 2. Draw a line of length 17 perpendicular to it. Connect the ends to make a right-angled triangle. The length of the hypotenuse of this triangle will be a non-integer.

Draw a line of length 17 perpendicular to the hypotenuse and make another right-angled triangle. Again the new hypotenuse will have a non-integer length. Repeat this until you get a hypotenuse of integer length. What is the length of this hypotenuse?

The sum of all the numbers in the eighth row of Pascal's triangle.

The smallest number whose sum of digits is 25.

The number of degrees in one internal angle of a regular polygon with 360 sides.

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 than can be made from the digits in red boxes.

A book has 386 pages. What do the page numbers on the two middle pages add up to?

In July, I posted the Combining Multiples puzzle.

Today's number is the largest number that cannot be written in the form $27a+17b$, where $a$ and $b$ are positive integers (or 0).