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.
I love wrapping presents. I’d like to say it’s because of the warm glow I have inside from giving a gift to someone else – and that feeling is certainly there to an extent – but I’m sorry to say the main reason is because I like the process of wrapping presents itself.
I like putting the present on the paper and making a judgement of how much paper to cut; I like using the scissors like a knife to cut a clean edge; I like folding the edge of the paper so that it looks nice and clean when you fold it over the present; I particularly like the part where you do the fold-in-the-sides-then-fold-up bit on the sides; and most of all I like the part where it’s all finished and your present is neatly encased in a piece of paper just the right shape with all the bits folded in neatly.
Yes, I know I’m weird.
But I reckon I’m not that weird. My daughter at 10 years old, still likes reciting the alphabet, though she learned to do this 6 years ago. My other daughter at 5 years old, will write her name over and over and over and over, seemingly getting pleasure out of the simple act. A musician will sometimes play a song they know well, for the sheer pleasure it, and almost any person will go up to a piano and play chopsticks. Many people I know like the experience of making scrambled eggs, no matter how many times they have done it before.
It seems that all people derive some pleasure in doing things well that they know how to do well, even though they have done it before. There is something about the repetition that gives you a sense of pleasure. Perhaps your brain likes to have the electrical signals pass down the well-worn paths where it’s not so much effort. Perhaps the experience helps you remember the buzz when you learned it for the first time.
I think perhaps the second reason is pretty accurate because I see myself doing it all the time in my work as well: guessing eigenvalues, calculating integrals, adding fractions and drawing conics. I love them all. I jump at the chance to do them with students in the MLC because I love doing them, no matter how many times I’ve done them before. And every time I do them, I remember with pleasure the first time I figured out how to do them myself.
But whatever the reason, I do get pleasure from doing the integral of ex cosh x or (cos x)2 or 1/(x2 – 1) – integrals I have done a hundred times – and it coming out to the answer it ought to. It’s the same pleasure I get from wrapping a present.
Sometimes you just enjoy doing something you know how to do.
Some years ago, I saw a snippet on the ABC science show Catalyst about insomnia – in particular, the flavour of insomnia where a person has trouble falling asleep at all. They reported on a trial study investigating the effectiveness of a tortuous new treatment for chronic insomnia. (You can find the published research here: Click here to go to insomnia article .)
The usual way to cure insomnia is to retrain your brain and your body to associate the bed with sleep rather than wakefulness. What they recommend is to only go to sleep when you’re really really tired, and if you don’t fall asleep in quarter of an hour, to get up and go to some other room until you feel tired enough to go to sleep again. Eventually, you’ll fall asleep in bed. Then you try again tomorrow night, and the next night, and the next night… Usually it takes a month.
The big problem with it is that people just don’t have the stamina to put themselves through all this for four weeks. Here’s where the radical treatment comes in: you compress the month of practice into 24 hours. The poor participant is put in a windowless room and practises going to sleep, and when they finally do fall asleep, they only get four minutes to sleep before they are woken up to try and fall asleep again. In this way you fit a month’s worth of falling-to-sleep practice in one day. Imagine how desperate you would have to be to sign up for this sort of thing!
Recently, it occurred to me that there are a lot of other skills that take a lot of practice to learn and this practice is usually drawn out over such a long period that people just don’t get through it all. One of these is statistics – in particular, the process of deciding which statistical procedures should be used to analyse your data.
In your standard stats course, the approach to teaching students to make decisions is to get them to do a project. This gives them practice at making decisions a grand total of once. And so students need a whole degree’s worth of projects, and probably years of working as a statistician, to learn how to make decisions. Hence, very few people ever get very good at making them. It’s just like the poor insomniac trying to cure their own insomnia once a night.
But what if you could, like the new insomnia treatment, compress all that practice into a short amount of time? What if you could pick out just the part where you make the decision and get students to make a lot of decisions all at once? Then they might get the necessary experience rather more quickly than the standard approach.
I tried it out last year with the med students. I gave them a quick lecture about how you make the decision of which hypothesis test to use. Then, I gave them 30 research questions and got them to make a decision for each one. They seemed to get the idea of how it worked. So much so that they actually had intelligent questions to ask afterwards!
I’m trying again this year, only this time the Medical School is letting me help design the whole stats teaching program, not just one lecture. Here’s hoping that a little bit of torture for a short time can alleviate months of pain later…
Theis comment was left on the original blog post:
Richard Knowling 27 January 2012: This is an awesome idea David! I only wish Mike Roberts had still been alive to hear about it!
