Puzzles
12 December
For a general election, the Advent isles are split into 650 constituencies. In each constituency, exactly 99 people vote: everyone votes for one of the two main parties: the Rum party or the
Land party. The party that receives the most votes in each constituency gets an MAP (Member of Advent Parliament) elected to parliament to represent that constituency.
In this year's election, exactly half of the 64350 total voters voted for the Rum party. What is the largest number of MAPs that the Rum party could have?
11 December
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 red digits.
| + | ÷ | = 2 | |||
| + | ÷ | ÷ | |||
| ÷ | ÷ | = 3 | |||
| ÷ | - | ÷ | |||
| ÷ | ÷ | = 1 | |||
| = 2 | = 1 | = 1 |
10 December
For all values of \(x\), the function \(f(x)=ax+b\) satisfies
$$8x-8-x^2\leqslant f(x)\leqslant x^2.$$
What is \(f(65)\)?
Edit: The left-hand quadratic originally said \(8-8x-x^2\). This was a typo and has now been corrected.
![](javascript:showlimage("advent2019quads.png"))
9 December
Arrange the digits 1-9 in a 3×3 square so that:
all the digits in the first row are odd;
all the digits in the second row are even;
all the digits in the third row are multiples of 3;
all the digits in the second column are (strictly) greater than 6;
all the digits in the third column are non-prime.
The number in the first column is today's number.
| all odd | |||
| all even | |||
| all multiples of 3 | |||
| today's number | all >6 | all non-prime |
8 December
Carol uses the digits from 0 to 9 (inclusive) exactly once each to write five 2-digit even numbers, then finds their sum. What is the largest number she could have obtained?
7 December
The sum of the coefficients in the expansion of \((x+1)^5\) is 32. Today's number is the sum of the coefficients in the expansion of \((2x+1)^5\).
6 December
Noel's grandchildren were in born in November in consecutive years. Each year for Christmas, Noel gives each of his grandchildren their age in pounds.
Last year, Noel gave his grandchildren a total of £208. How much will he give them in total this year?
5 December
28 points are spaced equally around the circumference of a circle. There are 3276 ways to pick three of these points.
The three picked points can be connected to form a triangle. Today's number is the number of these triangles that are isosceles.