Puzzles

In the annual tournament of Christmas puzzles, each player must play one puzzle match against each other player. Last year there were four entrants into the tournament (A, B, C, and D), and so 6 matches were played: A vs B, C vs D, A vs D, A vs C, D vs B, and finally B vs C.

This year, the tournament has grown in popularity and 22 players have entered. How many matches will be played this year?

The diagram to the right shows (two copies of) quadrilateral ABCD.

The sum of the angles ABC and BCD (green and blue in quadrilateral on the left) is 180°. The sum of the angles ABC and DAB (green and orange in quadrilateral on the left) is also 180°. In the diagram on the right, a point inside the quadrilateral has been used to draw two triangles.

The area of the quadrilateral is 850. What is the smallest that the total area of the two triangles could be?

120 is the smallest number with exactly 16 factors (including 1 and 120 itself).

What is the second smallest number with exactly 16 factors (including 1 and the number itself)?

Noel writes the integers from 1 to 1000 in a large triangle like this:

The number 12 is directly below the number 6. Which number is directly below the number 133?

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.

Noel writes the integers from 1 to 1000 in a large triangle like this:

The rightmost number in the row containing the number 6 is 9. What is the rightmost number in the row containing the number 300?

There are 3 even numbers between 3 and 9.

What is the only odd number \(n\) such that there are \(n\) even numbers between \(n\) and 729?

Holly draws a line of connected regular pentagons like this:

She continues the pattern until she has drawn 204 pentagons. The perimeter of each pentagon is 5. What is the perimeter of her line of pentagons?