This week I provided games and puzzles at a welcome lunch for new students in the Mathematical Sciences degree programs. I had big logic puzzles and maths toys and also a list of some of my eight most favourite puzzles on tables with paper tablecloths to write on.
One of the puzzles is the Seven Sticks puzzle, which I invented:
Seven Sticks I have seven sticks, all different lengths, all a whole number of centimetres long. I can tell the longest one is less than 30 cm long, because it’s shorter than a piece of paper, but other than that I have no way of measuring them. Whenever I pick three sticks from the pile, I find that I can’t ever make a triangle with them. How long is the shortest stick?
I sat down at one of the tables and I could see the students there were working on the Seven Sticks, so I asked them to explain what they had done. (WARNING! I will need to talk about their approach to the problem, so SPOILER ALERT!)
They told me about their reasoning concerning lengths that don’t form triangles, and what that will mean if you put the sticks in a list in order of size. They had used this reasoning to write out a couple of lists of sticks on the tablecloth which showed that a certain length of shortest stick was possible but that a longer shorter stick wasn’t possible. And so they knew how long the shortest stick actually was.
Only they said to me they hadn’t done it right.
I was surprised. “What?” I said. “All of your reasoning was correct and completely logical and you explained it all very clearly. Why don’t you think you’ve done it right?”
Their response was that what they had done wasn’t maths. They pointed across the table to what some other students were doing, which had all sorts of scribbling with calculations and algebra, and said, “See? That’s maths there and we didn’t do any of that.”
I told them that actually what they did was exactly what maths is – reasoning things out using the information you have and being able to be sure of your method and your answer. Just because there’s no symbols, it doesn’t mean it’s not maths.
Only later did I realise the implication of what had happened: these students are coming into a maths degree, which means they have done the highest level of maths at school, and they perceived that something was only maths if it had symbols in it. A plain ordinary logical argument wasn’t maths to them. And a scribbling of symbols was better than a logical argument, even if it didn’t actually produce a solution.
This made me really sad.
I wonder what sort of experiences they had that led them to this belief. I wonder what sort of experiences they missed out on that led them to this belief. I wonder how many other students have the same belief and I will never know. I wonder how I can help all the new students to see more in maths than calculations and symbol manipulation, and allow themselves to be proud of their work when they do it another way.
I created the Number Dress-Up Party puzzle way back in 2017 and every so often I stumble across it again when searching Twitter for other stuff. When I stumbled across it today, I decided it was time to write it up in a blog post.
The puzzle goes like this:
The Number Dress-Up Party
All the numbers have come to a dress-up party in full costume. They all know themselves which costume everyone else is wearing, but you don’t know.
If you pick any two of them and ask them to combine with +, -, × or ÷, they will point out which costume is the correct answer, and they’ll happily do it as often as you want. For example, you might ask for robot + bear and they will point to unicorn. (If the answer isn’t at the party, they’ll tell you that too.)
How do you correctly identify the numbers 0, 1, 2 and 3? How do you do it in as few steps as possible?
It is worth clarifying now that when I say “all” numbers I originally meant it to be all the real numbers. (But it is very interesting to think about how the problem is different or even possible if it means all the rational numbers or all the integers or all the natural numbers or some other set of numbers entirely.) It’s also worth pointing out that it is actually possible for you to guarantee that you actually find these numbers – you shouldn’t have the possibility of having to go through all the infinitely many costumes to be sure of finding 0, for example.
It’s also worth clarifying that the rules say you have to ask two different costumes to combine with an operation. If you can see how using two of the same costume might help you identify actual numbers, then you are thinking along some helpful lines. However, the puzzle is much harder and much more interesting if you have to use two different costumes every time.
I love this puzzle so much! The reason I love it is that it forces you to think about numbers and algebra in a completely different way to any other puzzle or problem I have seen. You really need to think about how numbers are related to each other and how they behave under operations in order to figure out a way to correctly identify these numbers from the way the costumes interact.
Also, in order to tell someone about your solution, or even figure it out in the first place, you need to find a way of talking about or writing about this, which is a lot more difficult than it might seem at first glance. I find it really interesting to see how people attack the problem of describing what they are doing in this problem.
The feel of this puzzle to me is the feel of an abstract pure maths course like abstract algebra or number theory or real analysis, where you are digging deep into how numbers work without reference to specific numbers per se. I would love to go into such a course and use this in the first lecture/tute to get students in the right sort of mindset to attack the rest of the course. I’d also love to do an extension to this as an investigation into how the Euclidean Algorithm for natural number division works. To all those people who haven’t had that sort of training I say you are doing what a pure mathematician does when you think about this problem! You never thought it would be so much fun, did you?
Anyway, there is the Number Dress-Up Party puzzle for posterity. There are several different solutions and they are all lovely. Have fun!
PS: If you feel like seeing how people have thought about this problem, and are ok with spoilers, then check out the replies to this tweet .
These comments were left on the original blog post:
mau 4 February 2019:
I did not see much of a difference between real and rational numbers: at least, the solution I devised is the same (and no real number can be generated in a finite number of steps, unless we stumbled into a costume specially related to that number: but maybe I am wrong). Integer and natural numbers need a different approach, indeed.
David Butler 11 February 2019:
Yes I didn’t see much of a difference between the real and rational number parties. Or indeed the complex number party either. I think once they’re closed under all four of the operations we’re allowed to use, you don’t get much extra. And yes I agree the natural numbers/integers are very interesting!
This post is about a puzzle I’ve been tweeting about for the last couple of days. I got it originally from a book I was given back in the 1980’s called “Ivan Moscovich’s Super Games”. In the book, Ivan calls this puzzle “Bits”, but I don’t think that’s nearly descriptive or cute enough, so I asked my daughter what it should be called and we have come up with the much better name of PANDA SQUARES.
Once upon a time, I did a PhD in projective geometry. It was all about objects called quadrals (a word I made up) – ovals, ovoids, conics, quadrics and their cones – and the lines associated with them – tangents, secants, external lines, generator lines. During the first two years, I did talks about my PhD research, which I could not resist calling “David Butler and the Philosopher’s Cone” and “David Butler and the Chamber of Secants”.
At that time, my use of JK Rowling’s titles had to stop because there was no suitable mathematical thing to insert into the third title. It’s been ten long years since “David Butler and the Chamber of Secants”, and finally I have found something to use. Hence, welcome to…
The “Four Fours” is a very well-known little problem that encourages some creative thinking and use of the order of operations. It goes like this:
Using exactly four of the number 4, any of the operations +, -, ×, ÷ and as many brackets as you like, see if you can produce all the natural numbers from 1 to 20.
Taking these instructions at face value, there are some numbers you simply can’t make with exactly four fours, and it’s rather interesting to watch how people deal with this. Mostly they come up with creative approaches to subvert the instructions, by allowing you to concatenate 4’s to make bigger numbers, or allowing a decimal point (shamefully without a zero in front of it). It worries me that no-one takes a more systematic approach and asks if they could produce all the missing ones with just one of these innovations, or asks exactly how many outcomes there are with the original operations only. But discussing what happens with the Four Fours is not the purpose of this post. The purpose of this post is to show you four fourfoursesque puzzles I’ve created which have encouraged some great learning.
Zero zeros
The Zero Zeros problem goes like this:
Using all five of the numbers 10, 100, 1000, 10 000, 100 000, any of the operations +, -, ×, ÷, and as many brackets as you like, make as many different numbers as you can whose digits contain no zeros.
I must confess I made this one up tonight in order to make my title have four alternatives rather than three. But it’s still, I think, a very interesting problem. It requires you to think about how zeros appear in the digits of a number when you do operations, and it is surprisingly difficult to come up with even one solution, letalone several (and yes there are several solutions).
UPDATE: We tried this at One Hundred Factorial today (the day after the post). Originally we used just the first four numbers 10, 100, 1000, 10 000, but every single person came up with the same first solution and then got stuck. Using five numbers seems to give simultaneously more scope for different answers and a more varied set of first answers. Either way, it was as interesting as predicted!
FURTHER UPDATE: I tried this in a Year 7 classroom, and a whole lot of interesting things happened! Check out the A Day of Maths series.
Only Ones
The Only Ones problem goes like this:
Using any of the operations of addition, subtraction, multiplication, division, and powers, as well as as many brackets as you want, and also as many of the number 1 as you need, make each of the numbers from 2 to 20. For example, here is a way to make 17:
\[(1+1+1)^{1+1}\times(1+1)-1=17\]
What is the smallest number of the number 1 needed to produce each of the natural numbers from 2 to 20? (Note: you can’t concatenate to make numbers like 11 — each 1 must stand alone as its own number.)
I find this one much more rewarding than the Four Fours. Firstly it allows for powers, which adds another operation to practice. (Though some people include this in the Four Fours to begin with.) Second, it requires a bit of working systematically in order to be sure you have the smallest number of 1’s possible. Thirdly it doesn’t create as much of a need to extend the operations allowed, because it’s quite interesting enough all by itself. Finally, it also has a nice extension problem which is to find out if there is a number whose digits are all 1, and can be calculated using the same number of 1’s as appear in it.
Here is a photo of Year 11 students working on this problem. When I suggested that perhaps they need to be more systematic, they decided to take a divide-and-conquer approach.
Pi on the floor
The Pi on the Floor problem goes like this:
Using any of the operations of +, -, ×, ÷, as many of the number π as you need, and as many of the floor function ⌊·⌋ as you need, make each of the whole numbers from 1 to 20. What is the least number of π’s required to make each number?
I made this one up on the spot at the One Hundred Factorial session on Pi Day (American Pi day anyway). I wanted something fourfoursesque that used the number pi, and it occurred to me that the floor function would allow us to create integers pretty easily. Little was I prepared for the interest it would generate or the clever maths that would be created by the students in response. I learned so very much about how the floor function worked, and about how close some multiples of pi were to various whole numbers.
Here is a photo of some people working on the problem. (I have blurred out the solutions though, so you can do them yourself!)
The i’s have it
The i’s Have It problem goes like this:
Using any of the operations +, -, ×, ÷, as many brackets as you want, and as many of the complex number number \(i\) as you need, make each of the numbers \(a+bi\) for \(a\), \(b = -2, -1, 0, 1, 2\).
How does your answer change if you are not allowed to use the – symbol?
We did this one at One Hundred Factorial in January 2016 and it was a most interesting problem. You had to really use the fact that \(i\) is an ordinary number that does actually do ordinary operations, especially to produce the real number results. Amie Albrecht suggested the extension problem of not being allowed to use the – sign, which forces you to use the fundamental property of \(i\) that \(i \times i = -1\), and very much changes the answer! Zero in particular is much harder to make without using minus!
Conclusion
So there you have it. Four alternatives to the Four Fours. I like them all better than the Four Fours, mostly because none of them imply that you can do something that’s impossible, and some actually challenge you to find the most efficient solution. Plus, they all make you think about functions or numbers or properties of number that you don’t think about that often. I and various friends, students and strangers have had a lot of fun thinking about them, and have learned a lot. Try them out yourself and tell me how they go!
UPDATE: I said earlier that discussing what happens when you do the Four Fours is not the goal of this post. But three years later I did discuss a way to modify the Four Fours to encourage particular mathematical behaviours.
Upon Amie Albrecht and Cathy Wilton’s request, I am writing a blog post about a problem we worked on at One Hundred Factorial recently. A few weeks ago we became interested in problems involving removing spots from dice.
At the end of last year, the MTBoS (Math(s) Twitter Blog-o-Sphere) introduced me to this very interesting task: you have a cross made of four equal squares, and you are supposed to colour in exactly 1/4 of the cross and justify why you know it’s a quarter. I call it “Quarter the Cross”.
