Advent calendar 2016

2016's Advent calendar ended with a murder mystery, with each of the murderer, motive, weapon and location being a digit from 1 to 9. Here are the clues:

Today's number is 191 more than one of the other answers and 100 less than another of the answers.

Today's number is the number of three digit numbers that are not three more than a multiple of 7.

Today's number is a palindrome. Today's number is also the number of palindromes between 111 and 11111 (including 111 and 11111).

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).

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 smaller number in a red box to the power of the larger number in a red box.

Here is a list of facts about today's number:

This year, I have spend a lot of time working on the AVE game engine. Today's answer is the code to the safe in this Christmas themed game.

The smallest number that is equal to the sum of its digits multiplied by ten more than the sum of its digits.

You start at A and are allowed to move either to the right or upwards.

How many different routes are there to get from A to B?

Today's number is the second smallest number that can be written as a×b×c×d×e×f×g×h×i, where a,b,...,i are all integers greater than 1.

Put the digits 1 to 9 (using each digit once) in the boxes so that the three digit numbers formed (reading left to right and top to bottom) have the desired properties written by their rows and columns.

When you add up the digits of a number, the result is called the digital sum.

How many different digital sums do the numbers from 1 to 10⁹¹ have?*

Today's number is the number of ways that 35 can be written as the sum of distinct numbers, with none of the numbers in the sum being divisible by 9.

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 digits in the red boxes.

What is the volume of the smallest cube inside which a rectangular-based pyramid of volume 266 will fit?

What is the maximum number of lines that can be formed by the intersection of 30 planes?

One of the digits of today's number was removed to leave a two digit number. This two digit number was added to today's number. The result was 619.