09 Jun My new favorite math game
As mentioned elsewhere, I like to play games in my math classes – and I find that, almost as fun as playing the games, is coming up with ideas for new ones.
I came up with the idea for my current favorite game a couple months ago. Jessie was looking for a game to play with Blake, and she mentioned it to me, thinking I might have an idea for one. She was more or less right, I had a list of several ideas that I thought he might enjoy, but I’d never gotten around to trying any of them out for myself. So, I told her I’d get back to her the next day, and then went home and checked my list.
The games I was thinking of all involved players combining numbers, using various arithmetical operations to achieve a defined goal – some of the ideas I was thinking of were things like:
players would try to combine five numbers (1 – 12) using addition, subtraction, multiplication or division to achieve the highest multiple of five, or the highest multiple of ten, or an even number closest to 100.
I had notes and ideas on a lot of variations of this same idea. The tricky part was that most of the possibilities I was considering have fairly obvious solutions – for instance if you need to achieve a high multiple of five, you simply need to do what you can to create any multiple of five, and then multiply that number by the remaining cards. That’s not so bad, there are challenges that present themselves, for instance there might be a few different solutions as to how you get that initial multiple of five, so the players would have to discern between them, but ultimately the game ended up being mostly just an exercise in multiplication – which is good, but not as interesting or challenging as what I imagined Blake might like.
So, I played around with it a bit, and eventually came up with a tentative solution. The breakthrough was to reverse the goal – I had been thinking in terms of achieving the highest outcome – combining the numbers to their maximum quantity – but that kept leading towards a minimal number of solutions and not a lot of creativity (and not much of a need for division). Some of these “find the highest number” games were pretty challenging and interesting, and I’ll probably use them from time to time in the future, but I wanted to do better. Finally, it occurred to me to aim for the lowest number, and then I knew I was onto something.
With the game targeting the lowest number, I found that there were many more possible solutions, some very simple, some very complex, some clunky, some elegant, and some tantalizingly elusive. Each group of cards presents a new combination of unknown possibilities – pretty fun!
So I came up with the following rules on my own.
- Five cards are placed in the middle of the table (each card depicts a number).
- Players have a timed period to try to combine the cards using the four basic arithmetical operations to achieve the lowest positive integer score.
So, with that much done, I called it a day, and looked forward to showing it to Jessie. As it turned out, I didn’t have a chance to see her the next day, and on Thursday when she was supposed to tutor with Blake, he was absent, and so the whole thing kind of got put off. Which was pretty lucky really, because the game needed some field testing before it was really ready to go.
Friday mornings, I tutor Jiyen, and thought he might be the perfect person to help me try out my new game. He and I started our session with the things we needed to cover, but got through everything we had to do with about twenty minutes to spare. I described the game to him and we tried it out. The first thing we discovered is that using timed intervals is no fun – there was too much pressure, and the interval was hard to anticipate – sometimes a solution would come right away, sometimes not. We eventually arrived at a sort of imprecise duration:
- If either player found a “zero,” they would declare it and show the other, ending that hand.
- If a player found a “one”, they could declare it, but play would continue until both players concluded that a “zero” was unachievable. At that point, any “ones” could be compared.
- In most instances, a “zero” or “one” is achievable, but there are times when a higher number is the best possible solution, in that case, play continues until both players are convinced that no better solutions are possible.
Jiyen and I tried some variations – using fewer than five cards and more than five cards, but my initial inclination towards five cards seemed pretty good – fewer made the options too limited, more made the possibilities too complex, we agreed that five was optimal.
With that done, I headed confidently into my Mental Math 2 class. I was anxious to try it out with multiple players. I had eight students in class. I explained the rules to them, uncertain whether they’d enjoy it or not, and uncertain whether it would even work with that number of students. It was interesting – most students loved it, a couple didn’t like it at all – it definitely challenged everybody. The best part – for me – came on the very first round. Somehow, in all the hands that Jiyen and I had played, neither of us noticed the flaw – that if you can create any zero with any two, three or four cards, you can then multiply it through to achieve a zero with all five. For instance, if you have two cards of the same number, or two cards that can be combined to the same number, then you can simply subtract them to get zero, and then multiply that zero times all of the remaining cards – this makes the zeroes a little too attainable. Gita spotted this the very first hand she played – thereby winning the hand, and supplying the game with its final rule.
So here are the Game Rules:
- Five cards are placed in the middle of the table.
- Players use paper and pencil to combine the numbers on the cards, using the four basic arithmetical operations to achieve the lowest non-negative integer score.
- Play continues until one of the players achieves a “zero” using each of the five cards, or until all players agree that no “zero” is possible.
- Players are not allowed to multiply by zero.
Here are a few sample games:
(8/4 + 7 + 11)/10 = 2
(10 – 8) – (11 – 4)/7 = 1
I don’t see a zero – can you find one?
(10 – 9) + 6/2 – 3 = 1
(10 – 9) – (6 – 3 – 2) = 0
This time I found one, can you find any others?
(11 x 3) – (3 x 8) – 7 = 2
8 /(11 – 7) – 3/3 = 1
This is another hard one. Do you see anything?
(12 + 1) – (5 + 5 + 2) = 1
(12/(5 + 5 + 2)) – 1 = 0
Sometimes, they’re pretty easy.
Blake and Jessie did finally play this, but I’m not sure it was as exciting for them as I’d hoped. Jiyen and I have continued exploring it though, and Dar and I have been playing it quite a bit. He’s been pushing me to try using higher number cards – which I’ve been reluctant to do, because it seemed like it would make the good solutions less attainable. The other day though, I finally let Jiyen talk me into trying it with the numbers up to 24, and it worked pretty well. I told him that I didn’t think it would work with numbers any higher than that though – but since then, I’ve been thinking that maybe it would – I just need to let go of the idea that “ones” and “zeroes” are the goal – it might be that with higher cards, the “ones” and “zeroes” will be less likely, but that might be ok, the other solutions might be even more varied and challenging – I’ll give it a try, next time Dar and I play, since he asked for it first.