# Christmas puzzles: the solutions

To celebrate Christmas this year we released a sequence of three linked puzzles on Christmas Eve, Christmas Day, and Boxing Day. If you haven’t had a chance, do give these puzzles a go! If you have tried these puzzles and would like to see the solutions, please read on.

## Puzzle #1: Christmas tree sudoku

If you haven’t tried solving a so-called thermo sudoku before, getting an intuition for how the thermometers work is key. Notice that if a $1$ lies on a thermometer it must lie on the bulb, and if a $9$ lies on a thermometer it must lie at the end. This gives a good way to start the puzzle, looking at the third row from the bottom where can a $1$ go? Once you are about a third of the way into this puzzle, it more-or-less turns into a normal sudoku and is relatively straightforward to complete. Contrary to normal practice when designing a sequence of puzzles like this, this was possibly the hardest of the three puzzles. That’s why we made it so that you could still solve puzzle #2 without solving this one. Continue reading

# Christmas puzzle #3: Colouring by numbers

Happy Boxing day! That means it’s time for the third and final Chalkdust Christmas puzzle. We hope you have been enjoying them so far! You can find the first two puzzles here and here.

## The rules

• Below is a 15×20 grid and each square contains a digit 0–9. Your job is to colour in each of the squares according to the rules below.
• If a square has already been coloured in as part of a previous rule, then it, together with the digit it contains, should be ignored—in other words you should apply the rules in the order they are given, and only to the remaining white squares.
• Numbers clued by a given rule may overlap, so a digit can be part of several answers corresponding to the same colour.
• Where a rule is of the form ‘Colour all numbers of type $x$ colour $y$’, the numbers will appear either horizontally left-to-right, or vertically top-to-bottom, never reversed or along diagonals.
• None of the rules refer to numbers which start with a 0.
• Use of Python, OEIS, Wikipedia, etc. is advised for some of the clues.

# Christmas puzzle #1: Christmas tree sudoku

Here at Chalkdust, we like to celebrate Christmas as much as the next magazine for the mathematically curious, and what better way to celebrate than with a few yuletide mathematical puzzles. We have three for you, the first one you can find below, the second one will be published tomorrow (Christmas Day), and the final one the day after (Boxing Day). They are the perfect accompaniment to an warming hot chocolate and mince pie. Each puzzle is related to the previous one, so keep a hold of your solutions ready for the next day. We hope you enjoy giving them a go and the whole team wishes you a very merry Christmas!

## The rules

• Normal sudoku rules apply: you must complete the 9×9 grid with the digits 1 to 9 such that each digit appears exactly once in each row, column, and 3×3 block.
• The digits that appear on each thermometer must strictly increase as you move away from the bulb. The colours of the thermometers are purely decorative and do not affect the puzzle.
• The digits on the baubles are all even.
• The digits on the stars are all prime.

# Can you solve these puzzles?

Issue 2 of Chalkdust magazine will be released on Tuesday 6th October, in just under 5 weeks’ time. While we work on writing, editing and designing the content for the magazine, here are a few puzzles to keep you entertained…

# Thoughts on the crossnumber

The deadline to enter the Chalkdust crossnumber #1 has now passed. The winners will be announced in next week’s blog post. In this blog post, Professor Kevin Buzzard shares some thoughts on the crossnumber.

In the rules we are told that there is a unique solution to the crossnumber. On the face of it this looks like an innocuous comment — a crossnumber for which this wasn’t true would be perhaps a little disappointing (or even unfair). However both existence and uniqueness of a solution to the crossnumber are not immediately obvious, and one has to hence decide what to do with this extra information. One could decide to verify it, by solving the puzzle and checking along the way that there is a unique solution. Alternatively, one could decide to use the information to help solve the puzzle! It is not clear to me if this is “cheating”. Let me give two examples to explain what I mean.