# 7 ways to tell if a number is divisible by 7

I’m sure you know how to check if a huge number is divisible by 3, 9, 11 or by powers of 2 like 2, 4, and 8; you were probably taught how to do this in primary school. However most of you were probably never taught how to test whether a number is divisible by 7. In this article we will explore seven different ways to do that.

7 is by far the least-loved single digit number when it comes to divisibility. Image: Chalkdust

Before we start, let’s introduce some notation which we’ll use throughout the article. An $n$-digit number $N$ in decimal base 10 is a number with digits belonging to the set $C_{10}=\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ and can be expressed as sum of powers of 10: $N=\sum_{i=0}^{n-1}10^ic_i,$ where  $c_i \in C_{10}$. We normally write this number by concatenating the digits using their position to indicate the corresponding power of 10: $c_{n-1}c_{n-2}\cdots c_1c_0$. The least significant digit of $N$ is $c_0$, and most significant digit is $c_{n-1}$. Continue reading

# 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.

# Revisiting the 1986 computer classic Number Munchers!

Ready to play Number Munchers. Image: ©MECC 1990, reproduced for the purpose of review.

If you were a child in the eighties or nineties, you might have seen the educational game Number Munchers on your school PC. It was originally released by MECC in 1986, and was re-released several times (for MS-DOS, Apple, and more). Nearly three decades later, Number Munchers received a Readers’ Choice Award in 2005 from Tech and Learning.

Believe it or not, I didn’t play it as a kid—rather I just watched a classmate play the 1990s version on a Macintosh. It wasn’t until two decades later (read: last winter) when I had a go at playing it. I couldn’t find the Macintosh version of the game, but I did come across the older MS-DOS version, so I played that.

## Yum, yum!

The controls are quite straightforward—just use the arrow keys to move your green muncher around, and the space bar when it’s time to eat a number. Granted, most games I have played are for the PC, so I find keyboard controls easy to use.

Your green guy is sitting in a 5 by 6 grid, and each square on the grid contains a number. You get points by eating numbers that satisfy the rule given on the top of the screen. Meanwhile, if you eat a wrong number, you lose one life. The game ends when you run out of lives. Example rules include:

• Multiples of 5: eat 5, 10, 15, etc
• Factors of 14: only eat 1, 2, 7 and 14
• Prime numbers: eat primes
• Equals 6: you get expressions such as $6\times 1$, $3 + 0$, and need to pick the ones that equal 6
• Less than 12: eat only the numbers 1–11

There’s even a challenge mode that lets you mix and match the rules! Moreover, there are lots of difficulty levels to pick from. There are 11 levels in total; they start at ‘third grade easy’ (that’s year 4 for Brits like me), and go all the way up to ‘seventh grade easy/advanced’, and finally eighth grade and above.

Number Munchers features five fearsome foes to fight or flee. Image: ©MECC 1990, reproduced for the purpose of review.

You will also want to avoid the Troggles—they are the monsters who want to eat your little muncher! It’s another surefire way to lose a life. When I first saw the game as a child, I didn’t notice that there were five types of Troggles, each coming in different colours and walking in specific patterns. I also forgot that when a Troggle walks over a square, it leaves a new number behind. If that’s not challenging enough, things start to get more frantic in later levels. More Troggles will turn up on the same board, and they’ll move faster, so you’d better be quick on your feet or have picked an easy maths mode! You’re also more likely to see what happens when Troggles meet: one eats the other, then the surviving Troggle continues walking as if nothing happened.

The Troggles at it again in this cutscene. Image: ©MECC 1990, reproduced for the purpose of review.

When you’ve eaten all the numbers on the board that fit the rule, you get to move on to the next level! Also, every three or four levels you get treated to a funny cutscene featuring the muncher and the Troggles! In most of the cutscenes, the Troggles try to capture the muncher, only for the plan to backfire, so the muncher gets the last laugh! You can even hear the muncher a little jingle, as if they were singing “Nyah-nyah-nyah-nyah-nyah-nyah!” Apparently there are at least five more fun cutscenes out there. No, not all of them feature Troggles. Sorry Troggle fans!

## My favourite mode

As a schoolgirl I watched my classmate play the level where you only eat prime numbers, and the moment he lost a life. No—he did not get eaten! The disaster was what he ate…the number 1. The game then said that 1 is not prime, but didn’t explain why.

Late breaking news from Number Munchers: 1 is not prime! Image: ©MECC 1990, reproduced for the purpose of review.

Then the teacher’s assistant was watching too. When the muncher lost a life, she turned to me and asked, “Why do you think the number 1 is not prime?”. How was I supposed to know? I was only just starting to learn what a prime number is! I was aware that a prime is divisible only by 1 and itself, but didn’t realise that these two divisors should be distinct. It only dawned on me years later, but I’d already moved into secondary school by then!

This is why the prime numbers round became my favourite level in the game. It showed me a something I didn’t realise until then, and made me go “ooh”. And now I’m older, I’m having no difficulties with the prime level…as long as there are no three-digit numbers!

## Overall

Number Munchers is definitely one of those maths games that can be enjoyed by people of (almost) all ages. Just make sure you didn’t pick the hardest difficulty setting! I did that, and I instantly regretted it—I found myself struggling to figure out which of the three-digit numbers I got were multiples of 19! It didn’t help that I initially misread the question, and thought I was supposed to avoid said multiples! An easy way to throw a life away. And as if I didn’t have enough to do already, I had to keep dodging the Troggles to make sure I didn’t eaten! Unsurprisingly I gave up, and switched to an easier setting.

We do not recommend starting with this mode. Image: ©MECC 1990, reproduced for the purpose of review.

If you’re after graphics, I recommend the 90s Apple version—the creatures are prettier in there (especially your little green muncher). The graphics on the DOS version are not as great, but the gameplay’s the same and the Troggles still look quite nice in that version. If you want to try the game yourself, the original version is available to the public on the Internet Archive, all for free. Better still, no emulator is required. What’s not to like?

Believe it or not, this is not the only maths-themed game in the Munchers series—there’s another game called Fraction Munchers! It features fractions instead of whole numbers, but I’ve never seen it! If you’ve been lucky enough to have played that game, why not send your review of Fraction Munchers to Chalkdust? It might just become an online article in here, too!

# Flo-maps fractograms: the game

This game corresponds to the article Flo-maps fractograms: the prequel. Have a go!

## Instructions

Here, you provide a numerator and denominator to receive a beautifully graphed pattern depending on the resultant decimal.

You must also designate the number of decimal places you wish to show: please do not use a number greater than 500.

Similarly, please try to keep the number of decimals a multiple of 100 for numbers larger than 100, and a multiple of 10 otherwise. This is not mandatory, however may result in visual bugs if not followed.

The recommended numbers of decimal places are: 10, 50, 100, 200, 500.

Please keep the denominator larger than the numerator, as the program will only illustrate decimal place patterns.

Once you have chosen your numerator, denominator, and number of decimal places, just press “Go”.

The “Superimpose” option will allow you to superimpose different patterns on top of each other. Please only use it after you have already shown one pattern.

# Flo-maps fractograms: the prequel

My original interest in decimal fractions was due to studying the ‘chaotic dropper’ experiment (see Fractograms from Chalkdust issue 02). Long before this, I had read about modelling the growth of a population and finding that it, too, can demonstrate chaotic behaviour. This is shown by using a logistic map. I realised the digits of decimal fractions could be subjected to the same modelling process. Let me explain.

# Rewatch the Issue 12 virtual launch event

Have you read through all of Issue 12 and feel lost without Chalkdust in your life? Then you are in luck because to celebrate Issue 12 in all its majesty, we hosted a virtual launch event YouTube livestream!

This featured not 1, not 2, not 4, not π, but 3 interactive workshops by the team behind Issue 12.

We figure out whether poetry is really just random nonsense or not, by creating Markov chain poetry and seeing how it measures up (check out this article for more on Markov chains, and this poem for inspiration).

We make our very own stunning mathematical drawings to colour in, using only a ruler and compass (like the ones on issue 12’s cover)—just the soothing activity you will need to take your mind off the fact that it’s another six months until Issue 13.

And last but certainly not least, we play some Countdown (numbers round, of course), presented by Nick Hewer and Rachel Riley (as soon as they return our calls).

Watch it again at youtu.be/egB0WJe-wPs. The launch event was live on Saturday 14 November, 2:00-3.30pm GMT.

# Crossnumber winners, issue 11

Hello everyone! It’s time to announce the winners of the Chalkdust prize crossnumber #11! Before we reveal the winners, here is the solution of the crossnumber.

 1 0 9 8 9 1 0 9 8 9 1 1 1 1 5 1 2 6 8 5 4 2 0 1 1 9 9 1 5 3 7 9 5 4 4 3 2 2 2 0 8 3 9 8 2 8 9 6 1 7 3 5 1 4 3 2 1 3 5 7 3 1 5 9 2 5 8 7 0 3 5 8 6 5 4 8 5 6 0 1 1 1 3 9 9 1 3 1 0 2 2 2 6 0 2 5 9 7 3 7 0 9 9 1 1 2 0 2 5 8 6 2 1 7 1 1 1 1 9 8 9 0 1 9 8 9 0 1

The sum of the across clues was 10994518584.

There were 74 entries, 65 of which were correct. The randomly selected winners are:

1. Paul Livesey, who wins a £100 Maths Gear goody bag,
2. Sam Dell, who wins a Chalkdust T-shirt,
3. Sarah Corbett, who wins a Chalkdust T-shirt,
4. George Panagopoulos, who wins a Chalkdust T-shirt.

Well done to Paul, Sam, Sarah and George, and thanks to everyone else who attempted the crossnumber. See you shortly in issue 12…