Negative numbers... why not?

[Graeme Cole]

Today on Countdown, one of the contestants accidentally said "eight minus ten" while giving her numbers solution. Rachel corrected her to "ten minus eight" and all was well. But this got me thinking about the rules of the numbers game.

You aren't allowed to use negative numbers in your working. The exact wording in the rules is "only positive integers may be used". But why is this? Being allowed to use negative numbers doesn't confer any advantage. Any method which uses negative numbers can easily be transformed into a similar method that doesn't. Is there anything fundamentally wrong with a contestant solving 75 2 1 9 3 10 -> 432 using 3 - 75 = -72, 2 + 1 - 9 = -6, -72 * -6 = 432?

Perhaps the rule is there to discourage showing off, but there are plenty of opportunities to do that in the numbers game anyway. And as with any way of showing off, whether it's using negative numbers, multiplying by 1, or using all six numbers when you only needed to use two*, if you mess it up, you throw away ten points and look like a prat. That's your own fault.

* As an aside, when the numbers game is ludicrously easy (e.g. 5 * 100 = 500), what I'd like to see occasionally is for Nick to say, before starting the clock, something like "you can both have the ten points, but just for fun, so we're not sitting around twiddling our thumbs for 30 seconds, see if you can find a way that uses all six numbers. Time starts now." Surely that's better than 30 seconds of nothing happening.

[Mark Deeks]

I remember Whiteley did that once when the target came up at exactly 100 and there was a 100 in the selection. fwiw.

[sean d]

I think enough viewers and contestants are intimidated by the numbers as it is, without some geeky college kid multiplying 2 minus numbers to get.... gasp... a positive number. I don't think there's any solution you can come up with that needs to go to minus numbers so no need to further alienate those who can't grasp, say, split multiplication, imo.

Now powers, roots and logs would make it interesting, and maybe we could throw in pi, e and i to shake it up.

[Guy Barry]

Any method which uses negative numbers can easily be transformed into a similar method that doesn't.

That seems to be a very good reason for disallowing their use. Why bring unnecessary complexity into the game?

The rule I do occasionally have a problem with is the one disallowing the use of fractions. Let's say the selection is 100 75 9 1 2 4 and the target is 954. One obvious way of doing it is (100 x (9 + (1/2))) + 4, but the rules don't permit it and there's no simple way of transforming it to a method that doesn't use fractions. (I remember seeing a contestant getting pulled up on something like this in the early days of the programme.) Is there any good reason for excluding such a solve?

* As an aside, when the numbers game is ludicrously easy (e.g. 5 * 100 = 500), what I'd like to see occasionally is for Nick to say, before starting the clock, something like "you can both have the ten points, but just for fun, so we're not sitting around twiddling our thumbs for 30 seconds, see if you can find a way that uses all six numbers. Time starts now." Surely that's better than 30 seconds of nothing happening.

Why not reprogram CECIL (or whatever it's called now) so that such targets don't come up? Any target that is the product (or sum) of two individual numbers would simply be eliminated.

[Zarte Siempre]

Let's say the selection is 100 75 9 1 2 4 and the target is 954. One obvious way of doing it is (100 x (9 + (1/2))) + 4, but the rules don't permit it and there's no simple way of transforming it to a method that doesn't use fractions.

What, like (100+4+2)*9 ?

[Guy Barry]

Let's say the selection is 100 75 9 1 2 4 and the target is 954. One obvious way of doing it is (100 x (9 + (1/2))) + 4, but the rules don't permit it and there's no simple way of transforming it to a method that doesn't use fractions.

What, like (100+4+2)*9 ?

That's not a transformation of the other method - it's a completely different method. I should have thought of an example that didn't have an obvious alternative solution. (I'm sure someone will.)

[Gavin Chipper]

But anyway the point with negative numbers is that they add no solving power and may add confusion so they are basically pointless. With fractions it would at least be a different argument.

[Matt Morrison]

Amen to that brother.

Here's a maybe-interesting question that seems definitely stupid at first but perhaps isn't: would allowing fractions make numbers games harder or easier?

[Guy Barry]

Here's a maybe-interesting question that seems definitely stupid at first but perhaps isn't: would allowing fractions make numbers games harder or easier?

I doubt whether contestants would want to use them very often, but they might occasionally make a difference.

One intriguing possibility is that they might come into play if it was possible to come closer to the target by declaring a fractional solution. Stephen Mangan actually did this in one of the 8 out of 10 Cats specials: set 50, 75, 8, 5, 1, 9 -> 896, he declared 894.44444... . At the time I thought it was just a silly joke but in fact it's equal to (75 * (8 + 5 - 1)) - (50/9), which I thought was rather impressive! As it happens, his opponents declared 895, so it didn't matter, but what if they'd been further away? Would he have been morally entitled to the points?

[Clive Brooker]

I should have thought of an example that didn't have an obvious alternative solution. (I'm sure someone will.)

Here are a couple:
3 3 10 8 9 6 : 955
25 10 10 5 4 1 : 343

[Gavin Chipper]

Amen to that brother.

Here's a maybe-interesting question that seems definitely stupid at first but perhaps isn't: would allowing fractions make numbers games harder or easier?

It can only make them easier to get them spot on, since you have everything you had before and more besides. But as for getting the closest possible to the target, which may be what you meant, then I don't know. I'd suspect probably harder.

[Clive Brooker]

To become fully proficient, you would have more tricks to learn. Therefore the game would be harder. No argument, surely?

[Graeme Cole]

But anyway the point with negative numbers is that they add no solving power and may add confusion so they are basically pointless. With fractions it would at least be a different argument.

I think it's an inertia thing. I don't think there's much argument for changing the rules to allow negative numbers, just as I don't think it would be worth bothering changing the rules to disallow them if they'd always been allowed. I was really just wondering why the rule against negative numbers existed in the first place - has it always been like that, or was the rule put in later on for some reason?

[Ciaran Thompson]

Strangely enough I was thinking of why can't you use negative numbers too but not because of the 8 - 10 thing. I thought of it just before that episode.

When working out a numbers game on one of the recent episodes, I found myself doing something like 7 - 9 + 5 to get a 3. You could of course say 7 - (9 - 5) but surely both are valid. Of course it's so rare (if ever) you need to go into negative numbers because the target is always 3 digits so you have to aim high as it were.

[Guy Barry]

When working out a numbers game on one of the recent episodes, I found myself doing something like 7 - 9 + 5 to get a 3. You could of course say 7 - (9 - 5) but surely both are valid.

You can simply rework that as 7 + 5 - 9.

[Ciaran Thompson]

When working out a numbers game on one of the recent episodes, I found myself doing something like 7 - 9 + 5 to get a 3. You could of course say 7 - (9 - 5) but surely both are valid.

You can simply rework that as 7 + 5 - 9.

Yeah I know. That's yet another way which I use often, but for some reason I found myself going into negatives that time. I can understand why negative numbers are not allowed, so as not to complicate the game or confuse the viewers, who many find the numbers games quite daunting as it is. Yet I think people are generally more familiar with negative numbers than how and when to use brackets in their working, due to the fact we're all so familiar with the temperature going below zero and how the years count backwards like negative numbers in BC. Also, as I recall from my school days, you learn about negative numbers at a younger age than you do about brackets, as presumably they are easier to grasp.