Puzzles

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?

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

what are \(I\), \(V\) and \(X\)?

2018's Advent calendar ended with a logic puzzle: It's nearly Christmas and something terrible has happened: one of Santa's five helpers—Kip Urples, Meg Reeny, Jo Ranger, Bob Luey, and Fred Metcalfe—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:

1,0,2,0,1,1

The sequence of six numbers above has two properties:

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

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

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.

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.

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