# Puzzles

## Coloured weights

You have six weights. Two of them are red, two are blue, two are green. One weight of each colour is heavier than the other; the three heavy weights all weigh the same, and the three lighter weights also weigh the same.

Using a scale twice, can you split the weights into two sets by weight?

## Not Roman numerals

The letters \(I\), \(V\) and \(X\) each represent a different digit from 1 to 9. If

$$VI\times X=VVV,$$
what are \(I\), \(V\) and \(X\)?

## Advent 2018 Logic Puzzle

2018's Advent calendar ended with a logic puzzle: It's nearly Christmas and something terrible has happened: one of Santa's five helpers—Meg Reeny, Bob Luey, Jo Ranger, Fred Metcalfe, and Kip Urples—has stolen all the presents during the North Pole's annual Sevenstival. You need to find the culprit before Christmas is ruined for everyone.

Every year in late November, Santa is called away from the North Pole for a ten hour meeting in which a judgemental group of elders decide who has been good and who has been naughty. While Santa is away, it is traditional for his helpers celebrate Sevenstival.
Sevenstival gets in name from the requirement that every helper must take part in exactly seven activities during the celebration; this year's
available activities were billiards, curling, having lunch, solving maths puzzles, table tennis, skiing, chess, climbing and ice skating.

Each activity must be completed in one solid block: it is forbidden to spend some time doing an activity, take a break to do something else then return to the first activity.
This year's Sevenstival took place between 0:00 and 10:00 (North Pole standard time).

During this year's Sevenstival, one of Santa's helpers seven activities included stealing all the presents from Santa's workshop.
Santa's helpers have 24 pieces of information to give to you, but the culprit is going to lie about everything in an attempt to confuse you, so be careful who you trust.

Here are the clues:

1

Meg says: "Between

Meg says: "Between

**2:33**and curling, I played billiards with Jo."15

Kip says: "The curling match lasted

Kip says: "The curling match lasted

**323**mins."24

Fred says: "In total, Jo and Meg spent

Fred says: "In total, Jo and Meg spent

**1**hour and**57**mins having lunch."8

Meg says: "A total of

Meg says: "A total of

**691**mins were spent solving maths puzzles."17

Jo says: "I played table tennis with Fred and Meg for

Jo says: "I played table tennis with Fred and Meg for

**2**+**8**+**5**mins."23

Meg says: "

Meg says: "

**1:32**was during my 83 min ski"7

Meg says: "The number of mins the curling game lasted is a factor of

Meg says: "The number of mins the curling game lasted is a factor of

**969**."16

Jo says: "I started skiing with Bob, and finished before Bob at

Jo says: "I started skiing with Bob, and finished before Bob at

**8:45**."5

Jo says: "At

Jo says: "At

**4:45**, Fred, Bob, Kip and I started a curling match."14

Fred says: "I spent

Fred says: "I spent

**135**mins playing chess with Meg."20

Meg says: "Jo started skiing at

Meg says: "Jo started skiing at

**7:30**."4

Bob says: "I went for a

Bob says: "I went for a

**150**min ski."13

Kip says: "Jo started skiing at

Kip says: "Jo started skiing at

**6:08**."22

Fred says: "Bob, Kip and I finished lunch at

Fred says: "Bob, Kip and I finished lunch at

**3:30**."6

Bob says: "I played billiards with Kip from 0:00 until

Bob says: "I played billiards with Kip from 0:00 until

**1:21**."12

Fred says: "Between 3:30 and 4:45, there were

Fred says: "Between 3:30 and 4:45, there were

**3**people climbing."21

Fred says: "In total, Bob, Meg and I spent

Fred says: "In total, Bob, Meg and I spent

**269**mins ice skating."10

Meg says: "Between 0:00 and

Meg says: "Between 0:00 and

**1:10**, I was ice skating."19

Jo says: "At

Jo says: "At

**1:12**, Fred and I were both in the middle of maths puzzles."3

Jo says: "Straight after curling, I had a

Jo says: "Straight after curling, I had a

**108**min game of chess with Kip."9

Fred says: "At

Fred says: "At

**2:52**, I started having lunch with Bob and Kip."18

Jo says: "I spent

Jo says: "I spent

**153**mins solving maths puzzles."2

Fred says: "I was solving maths puzzles for

Fred says: "I was solving maths puzzles for

**172**mins."11

Meg says: "I spent

Meg says: "I spent

**108**mins solving maths puzzles with Bob."## 24 December

1,0,2,0,1,1

The sequence of six numbers above has two properties:

- Each number is either 0, 1 or 2.
- Each pair of consecutive numbers adds to (strictly) less than 3.

Today's number is the number of sequences of six numbers with these two properties

## 23 December

Today's number is the area of the largest area rectangle with perimeter 46 and whose sides are all integer length.

## 22 December

In base 2, 1/24 is
0.0000101010101010101010101010...

In base 3, 1/24 is
0.0010101010101010101010101010...

In base 4, 1/24 is
0.0022222222222222222222222222...

In base 5, 1/24 is
0.0101010101010101010101010101...

In base 6, 1/24 is
0.013.

Therefore base 6 is the lowest base in which 1/24 has a finite number of digits.

Today's number is the smallest base in which 1/10890 has a finite number of digits.

Note: 1/24 always represents 1 divided by twenty-four (ie the 24 is written in decimal).

## 21 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 smallest number you can make using the digits in the red boxes.

+ | ÷ | = 2 | |||

× | + | - | |||

× | - | = 31 | |||

+ | + | - | |||

- | × | = 42 | |||

= 37 | = 13 | = -2 |

## 20 December

Today's number is the sum of all the numbers less than 40 that are not factors of 40.