Music Puzzles

So during one of my many, many random thoughts, something occurred to me. As a big fan of maths puzzles (remember this post?) I pondered the idea of transforming a maths puzzle into a music one. After all, anyone who knows almost nothing about music knows that it’s pretty much maths with sound. The majority of maths puzzles are based on a square grid and require you to input symbols that represent quantities. ie 1, 2, 3, 4.

Well funnily enough, musical notes are exactly that. Symbols that represent quantities. Portions of a beat, or of a note, or … well forget the preferred definition. Below are some examples of how you could easily transfer the essence of a maths puzzle into a music puzzle.

I present to you… Musical KenKen… or should that be… KenKen Beatz. Ugh, no, the first one.

*edit: Surely MusiKenKen??

First, here’s the available notes for each puzzle…Technically a crotchet (the third one down) is a ‘quarter note’, but I’m pretty sure we can say it lasts one beat. My music notation isn’t THAT rusty. Just… 15 years since I last thought about it.

note key

puz1

So this is puzzle one. Fill in each coloured grid with the correct notes to add up to the total amount of beats provided. No notes can be repeated on rows or columns.

Figured it out? Scroll down.

puz1ans

Puzzle two lets you have an EXTRA NOTE at your disposal… the semi quaver!

puz2

and the answer is below:

puz2ans

And finally puzzle 3, which allows you a fifth note, the mighty semibreve:

puz3

Cripes!

Answer below:

puz3ans

You could make them easier by providing a few notes here and there. You could also make them harder by introducing rests or dotted notes, or demisemiquavers etc, or different ‘operations’ like “-1″ meaning ‘the difference between the lengths of the two notes is 1 beat.”

Fun! Why do I occupy my brain with such random things? Who knows.

Pretty sure you could do this with a few other maths puzzles too… hmmm…

edit* Futoshiki would barely need altering:

v1 v2

Pose a Puzzle

Ah we all love a good puzzle. I’ve spoken a few times about my love of them, and it’s kind of inferred by almost everything on this website.

There are some great maths puzzles. Not only do they excite students in a way that perhaps no textbook page can, they also make you think. I don’t think any good puzzles can be done just by applying a mindless algorithm to a familiar looking structure.

I could write a billion words on puzzles so I’m going to have to be more specific. In my mind there are two kinds of maths puzzles.

Type 1 is the ‘one off’ puzzle. These can be presented, solved and moved to one side. Not that they aren’t great, they’re FANTASTIC, i write them all the time on this very website. Here are a couple of other great examples of this kind of puzzle:

B7FDCZ6IgAAo7ai

This one is by Chris Smith and is really, really good.

And Stephen Cavadino has curated some other great examples on his blog here.

But for this post I’m going to concentrate on the second type : the reusable puzzle. These are the sort that can be adjusted and readjusted time and time again for almost limitless variation. Think Sudoku but maths.

So here are some of my favourites:

1. KenKen

kenken-af

These look and feel very similar to Sudoku, but aren’t. They require calculations and foresight to solve. A solved version is below:

250px-KenKenSolution.svg

As with most maths puzzles, you are confined to using the numbers 1 to x (x being the length and usually also height of the grid) with no repetition on rows or columns. Each sub-grid has a rule attached, such as 2+ (meaning must sum to 2) or -3 (meaning must have a difference of 3). The rest is fairly self explanatory. These are great for reasoning, and practising number operations. They get incredibly difficult as you increase the size of the grid too. There’s a KenKen generator (with solutions) here. Or you could challenge yourself with this blank 5×5 in the meantime:

5x5-c2a189f02b341432475258003eddcb8d

2. Futoshiki

p1

Again, the repetition rule applies with this puzzle, and you use 1-4  for a 4×4 puzzle etc.

Essentially you must abide by the inequalities, which force a single solution. Funnily enough, I always use them when I’m teaching inequalities. Also any old starter. The solution is below if you can bare it no longer. Again, here is a generator with solutions.

p1s

3. Kakuro

Hosted by the same site is the Kakuro puzzle (generator link)

p2

You may have seen these in the newspaper. Here you have to use digits 1 – 9 and ensure each row / column total is equal to the number provided at it’s beginning.You are not allowed to use the same digit more than once to obtain a given sum. Again, increase the grid size, increase the difficulty. Good to practise addition skills.

Solution below.

p2s

4. Shikaku

These puzzles aren’t ALL Japanese, but most of them are. Shikaku puzzles are essentially area / factor pair puzzles.

shik

You need to draw squares / rectangles (but no other shapes like L-Shapes etc) so that each shape contains only one number, and that number is the area of the shape. There is a generator here, but sadly no answers this time!! I did this one all by myself (*so* proud…) just to show you how they work:

sol

5. Math Squares

ms1

Mercifully the generator gives solutions.

Back to the 1-9 rule here for 9 blank squares. These get exponentially difficult as you increase the grid size, you have been warned! Great for practising number operations, but also negative numbers.

ms

6. SkyScrapers

These are possibly my favourite, but quite hard for students to get their heads around initially.

ss

So we’re using 1-4, same old rules. But here, each number represents the *height* of a skyscraper, so 4 is ‘taller’ than 3 etc. The numbers provided on the grid show how many skyscrapers someone standing at that point would be able to see in that direction. So if it says ‘1’, they can only see 1 skyscraper, hence the first number would be ‘4’, the tallest skyscraper (blocking the view of the other ones). Similarly, if the number by the arrow is ‘4’, then the sequence must be 1, 2, 3, 4 as each skyscraper can be seen behind the preceding one. It’s clever and I love how it forces you to visualise numbers differently. Generator (with answers) here. Solution below.

ss1

They are also clever at making them harder too. Not just bigger grids…

sss

7. The Daily SET Puzzle

If you’re unfamiliar with ‘The SET Puzzle‘, fix that now! It’s a great logic puzzle that builds knowledge and understanding of sequences and patterns. It’s even advocated by Numberphile peeps. I have a physical set that I play with my son, although the game itself isn’t as good as the daily puzzle, so we end up just setting the cards out like the online puzzle and doing it that way.

set

Here you must create six unique ‘sets’ using only the cards provided. Each set must be either: same colour same shape different pattern, or different colour, same shape, same pattern, or same colour, different shape, same pattern, or EVERYTHING different. Got that? Didn’t think so. You’ll get used to it. Solutions are posted the day after.

It changes screen as soon as you get all 6, so here’s 5 solutions to help you get your head around it:

ans

8. Find The Factors

I’ve talked about these great puzzles before. They are the best way I’ve found to get students of *any* ability to practise multiplication tables. Essentially you have a 10×10 or 12×12 grid with a jumbled up times table grid on it, with lots missing. Piece together the clues provided to complete the table. Students could just fill in the column and row headings (the tricky bit) or the whole thing if their multiplication is a bit rubbish generally. Level Six puzzles are HARD.

345-puzzle

These took me a while to understand, so I’ll start you  a solution below:

ffs

Solutions are posted regularly. Site is here.

9. Cryptarithms

These are also really, really cool. Sometimes listed as Alphametrics. Make addition challenging and interesting again for students who got bored of it long ago. Essentially swap digits for letters (although be careful to distinguish this from algebra!!). The cleverest puzzles make sentences / words (solution on the right):

cr

It’s quite hard to find a nice supply of these things. There are some great ones (without answers!) over at Don Steward’s amazing website. and a handful here and here. That last link has a generator but I’ve found it to be a bit dodgy.

10. 7Puzzle

http://7puzzleblog.com/ has a huge array of puzzles based around playing with a small group of numbers to arrive at either a solution, or as many solutions as possible, or a variety of solutions using numbers in a certain order … or… oh there’s so many variations. It’s fantastic. I use this site for starters every week without fail. Created by the brilliant Paul Godding. Here’s one example:

pg

11. Balance Problems

photo-31

Again, featured here before. These are short in supply, but good to solve algebraically if needed. Order each shape according to its weight. Once more, a nice selection at Don Steward, but they seem to have originated here.

So that’s it. I have collated a range of puzzles as booklets (a full school year’s worth) for use as starters available on TES here. Enjoy!