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**2019-01-01**

It's 2019, and the Advent calendar has disappeared, so it's time to reveal the answers and annouce the winners.
But first, some good news: with your help, Santa was able to work out who had stolen the presents and save Christmas:

Now that the competition is over, the questions and all the answers can be found here.
Before announcing the winners, I'm going to go through some of my favourite puzzles from the calendar, reveal the solution and a couple of notes and Easter eggs.

### Highlights

My first highlight is the first puzzle in the calendar. This is one of my favourites as it has a pleasingly neat solution involving a
surprise appearance of a very famous sequence.

### 1 December

There are 5 ways to write 4 as the sum of 1s and 2s:

- 1+1+1+1
- 2+1+1
- 1+2+1
- 1+1+2
- 2+2

Today's number is the number of ways you can write 12 as the sum of 1s and 2s.

My next highlight is a puzzle that I was particularly proud of cooking up: again, this puzzle at first glance seems like it'll take a
lot of brute force to solve, but has a surprisingly neat solution.

### 10 December

The equation \(x^2+1512x+414720=0\) has two integer solutions.

Today's number is the number of (positive or negative) integers \(b\) such that \(x^2+bx+414720=0\) has two integer solutions.

My next highlight is a geometry problem that appears to be about polygons, but actually it's about a secret circle.

### 12 December

There are 2600 different ways to pick three vertices of a regular 26-sided shape. Sometime the three vertices you pick form a right angled triangle.

Today's number is the number of different ways to pick three vertices of a regular 26-sided shape so that the three vertices make a right angled triangle.

My final highlight is a puzzle about the expansion of a fraction in different bases.

### 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).

### Notes and Easter eggs

I had a lot of fun this year coming up with the names for the possible theives. In order to sensibly colour code each suspect's clues,
there is a name of a colour hidden within each name:
F

**red**Metcalfe, J**o Range**r, Bo**b Lue**y, Me**g Reen**y, and Ki**p Urple**s. Fred Metcalfe's colour is contained entirely within his forename, so you may be wondering where his surname came from. His surname is shared with Paul Metcalfe—the real name of a captain whose codename was a certain shade of red.On 20 December, Elijah Kuhn emailed me to point out that it was possible to solve the final puzzle a few days early: although he could not yet work
out the full details of everyone's timetable, he had enough information to correctly work out who the culprit was and between which times the theft
had taken place.

Once you've entered 24 answers, the calendar checks these and tells you how many are correct. This year, I logged the answers that were sent
for checking and have looked at these to see which puzzles were the most and least commonly incorrect. The bar chart below shows the total number
of incorrect attempts at each question.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

Day |

You can see that the most difficult puzzles were those on
13,
24, and
10 December;
and the easiest puzzles were on
5,
23,
11, and
15 December.

I also snuck a small Easter egg into the door arrangement: the doors were arranged to make a magic square, with each row and column, plus the two diagonals, adding to 55.

The solutions to all the individual puzzles can be found here. Using the clues, you can work out that everyone's seven activities
formed the following timetable.

Bob Luey | Fred Metcalfe | Jo Ranger | Kip Urples | Meg Reeny |

0:00–1:21 Billiards | 0:00–2:52 Maths puzzles | 0:00–2:33 Maths puzzles | 0:00–1:21 Billiards | 0:00–1:10 Ice skating |

1:10–2:33 Skiing | ||||

1:21–2:52 Ice skating | 1:21–2:52Stealing presents | |||

2:33–4:45 Billiards | 2:33–4:45 Billiards | |||

2:52–3:30 Lunch | 2:52–3:30 Lunch | 2:52–3:30 Lunch | ||

3:30–4:45 Climbing | 3:30–4:45 Climbing | 3:30–4:45 Climbing | ||

4:45–5:42 Curling | 4:45–5:42 Curling | 4:45–5:42 Curling | 4:45–5:42 Curling | 4:45–5:42 Lunch |

5:42–7:30 Maths puzzles | 5:42–7:30 Ice skating | 5:42–7:30 Chess | 5:42–7:30 Chess | 5:42–7:30 Maths puzzles |

7:30–10:00 Skiing | 7:30–9:45 Chess | 7:30–8:45 Skiing | 7:30–10:00 Maths puzzles | 7:30–9:45 Chess |

8:45–9:45 Lunch | ||||

9:45–10:00 Table tennis | 9:45–10:00 Table tennis | 9:45–10:00 Table tennis | ||

Following your investigation, Santa found all the presents hidden under Kip Urples's bed, fired Kip and sucessfully delivered all the presents
on Christmas Eve.

### The winners

And finally (and maybe most importantly), on to the winners: 73 people submitted answers to the final logic puzzle. Their (very) approximate locations are shown on this map:

From the correct answers, the following 10 winners were selected:

1 | Sarah Brook |

2 | Mihai Zsisku |

3 | Bhavik Mehta |

4 | Peter Byrne |

5 | Martin Harris |

6 | Gert-Jan de Vries |

7 | Lyra |

8 | James O'Driscoll |

9 | Harry Poole |

10 | Albert Wood |

Congratulations! Your prizes will be on their way shortly. Additionally, well done to
Alan Buck, Alex Ayres, Alex Bolton, Alex Lam, Alexander Ignatenkov, Alexandra Seceleanu, Andrew Turner, Ashwin Agarwal,
Becky Russell, Ben Reiniger, Brennan Dolson,
Carl Westerlund, Cheng Wai Koo, Christopher Embrey, Corbin Groothuis,
Dan Whitman, David, David Ault, David Kendel, Dennis Oltmanns,
Elijah Kuhn, Eric, Eric Kolbusz, Evan Louis Robinson,
Felix Breton, Fred Verheul,
Gregory Loges,
Hannah,
Jean-Noël Monette, Jessica Marsh, Joe Gage, Jon Palin, Jonathan Winfield,
Kai Lam,
Louis de Mendonca,
M Oostrom, Martine Vijn Nome, Matt Hutton, Matthew S, Matthew Wales, Michael DeLyser, MikeKim,
Naomi Bowler,
Pranshu Gaba,
Rachel Bentley, Raymond Arndorfer, Rick Simineo, Roni, Rosie Paterson,
Sam Hartburn, Scott, Sheby, Shivanshi, Stephen Cappella, Steve Paget,
Thomas Smith, Tony Mann,
Valentin Vălciu, Yasha Ayyari, Zack Wolske, and Zoe Griffiths, who all also submitted the correct answer but were too unlucky to win prizes
this time.

See you all next December, when the Advent calendar will return.

### Similar posts

Christmas card 2018 | Christmas (2018) is coming! | Christmas (2017) is over | Christmas card 2017 |

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