The Maths of Pointless

How often do we come up against people who are convinced that mathematics is pointless? TV game show Pointless requires contestants to obtain the lowest score to win against their opponents. Plus magazine discusses the maths behind this idea in two articles. The first looks at luck and unluckiness in TV game shows (with reference to Pointless); the second looks at some of the more general mathematical ideas related to Pointless specifically.

1. Pointless: The maths of TV gameshows

2. A Pointless return!

What's my score?

I hadn't played Yahtzee in years, but the other night I settled down to possibly the best couple of games I've ever played.

For those not in the know, Yahtzee is a game from Hasbro (My copy's from MB Games... that's how long it's been!) that's been around since 1956. Put simply, it's a game in which players roll five dice and are given a score based on what is rolled.

We played two games, and they were probably my best ever scores (though that doesn't say much; I'm not exactly a pro). In my first game, I scored exactly 80% of the maximum possible mark (assuming no bonuses for extra "yahtzee*"s), and in the second I scored 89% of the maximum, rounded to the nearest whole

1. What was my score in the first game?
2. In the second game, what's the highest score I could have got? What's the lowest?

Have a go at the questions before looking at the answers below!

The solutions

I'll remind you here as well that there are usually many, many ways of solving problems, so if you haven't done it the same way as me that doesn't mean that you've done it wrong. In fact, if you do it a different way I'd like to see an explanation in the comments- it might even be a better** way of doing it!

Part 1)

Look at the information I've given you: I got 80% of the total possible score. Assuming you know something about percentages*** then all you need to know is the total possible score. If you don't know the game, here's a scorecard to look at:

You just need to think of the best possible score for each roll, and record it, then add anything up, taking into account any bonuses. The overall total should come to 375. Comment and ask if you're not sure how to get that.

Now all you need to do is work out 80% of 375, then you've got my score for game 1:

One way of doing this is to find 10% (divide by 10) and then multiply by 8 (as 8 lots of 10% is 80%). So:

375 / 10 = 37.5

37.5 * 8 = 300

• My score for game 1 was 300 points exactly!

Part 2)

This one's trickier. The question asks for my highest and lowest possible scores because I didn't get exactly 89% - I rounded the answer. This means that finding 89% of the maximum possible score won't give the score I got, as it could have been a bit smaller or a bit bigger than that. In fact, if you work out 89% of 375...

375 / 100 * 89 = 333.75

... you don't even get a whole number, so it's actually impossible to get exactly 89% of the maximum score in a game of Yahtzee. So we'd have to make a guess. Luckily we can use the information given to make that guess as accurate as possible- we can find a sensible low- end for our guess, and a sensible high- end, and go somewhere in between. That's what the question means when it asks for the highest and lowest scores that I "could have" got.

To do this we need to look at that percentage first: I said I rounded it to the nearest whole percent, which was 89. So what are the lowest and highest answers I could have got before I rounded it? Any more than 89.5%, and I'd have rounded it up to 90%. Any less than 88.5%, and it'd have gone down to 88%. If we work out the two percentages in bold, we'll have our answers:

375 / 100 * 88.5 = 331.875

375 / 100 * 89.5 = 335.625

They're the highest and lowest scores, but now our common sense should kick in: there's no way of getting a decimal score in Yahtzee! That means the for the lowest score, we've got to pick 331 or 332, and for the highest, we must choose 335 or 336. But which one? Well, the lowest score that would end up being rounded to 89% would be 331.875. Any lower than that and we won't get 89%, so it must be higher than that: we pick 332. Using similar reasoning, but at the other end of the scale, we must pick 335 as the highest score that I could have got.

• My score for game 2 was somewhere between 332 and 335 inclusive.

Is it possible to be any more accurate than this?





* In the game Yahtzee, a 'yahtzee' is rolling five of the same number in one turn, i.e. after three rolls you end up with 5 twos, or 5 sixes, etc.
** What makes one answer 'better' than another, assuming that both are correct? That might be a subject for another post...
** If you don't, ask me for a post ;-)