Between twelve and five

Two months ago or so I spent some hours at the Hodges Figgis (a bookshop I love, which is on Dawson Street, here in Dublin). I love to go to the second floor and leaf through the books of the scientific section. It is the closest thing to the Casa del Libro in Madrid. There I found a book, The Maths Gene, by Keith Devlin. On page 19 (of that edition) there is the following test:

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You have to answer as quickly as you can:
1 - 1 = ?
4 - 1 = ?
8 - 7 = ?
15 - 12 = ?

And now, quickly, choose a number between 12 and 5!

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Supposedly you have chosen seven (the full story can be found also here). I can't remember now what did I choose, but it wasn't seven. So the first thing I thought was "hm, this is not quite serious". But of course, you cannot say something like that and believe it straight away. An experiment is needed... hooray!

The idea: Let's test this idea:

When you ask somebody to make subtractions and then you ask him/her for a number between two numbers, he/she unconsciously keeps subtracting.

The "material": 36 people took part in my experiment (all of them friends, relatives and colleagues).

The set-up: To do this experiment, I spent one week asking people I know for subtractions and numbers. Obviously, I didn't tell them anything beforehand, or otherwise they would have been conditioned to give an "interesting" answer. I always asked them in the office corridors, at the end of a phone call and so. This way, they didn't have a lot of time to think.

Firstly I considered two groups, twelve people each:

Group A: I made the experiment exactly like in Keith Devlin's book, with the same subtractions to do.

Group B: I told them "give me a number between 12 and 5" (not asking them for any subtractions).

The measures: I made some mistakes, like asking for a number between 7 and 12, but obviously I excluded these cases.

Here are the results for the group A:

answer 5 --> 0 people.
answer 6 --> 2 people.
answer 7 --> 7 people.
answer 8 --> 1 people.
answer 9 --> 1 people.
answer 10 --> 0 people.
answer 11 --> 0 people.
answer 12 --> 1 people.

And here are the results for the group B:

answer 5 --> 1 people.
answer 6 --> 1 people.
answer 7 --> 6 people.
answer 8 --> 2 people.
answer 9 --> 2 people.
answer 10 --> 0 people.
answer 11 --> 0 people.
answer 12 --> 0 people.

May be it is clearer with a couple of graphs:

Group A:


OK, maybe it is not a 90% of people, but slightly more than one half picked up seven. This was surprising for me.

Group B:


I would say both graphs are very similar! With subtractions seven people answered seven, and without subtractions it was six people. Yes, if you say we cannot expect great statistics out of twelve people you are right. But anyway, this is indicative that asking for subtractions is not the key point here.

What is the real reason? I would say lots of people like number seven, the "lucky number", and would have answered that even if I had asked for a number between 1 and 100. There is a group of "fans of number seven", but how large is this group?

What if the main point is the order we ask for the numbers? I kept asking all the time for a number "between twelve and five", but it would be more natural asking for a number "between five and twelve". So, I extended the experiment and considered a third group:

Group C: I told them "give me a number between 5 and 12" (in this more natural order, not asking for any subtractions).

This sub-experiment was to check the following sub-idea:

If you ask for a number "between twelve and five" most of the times you get seven; if you ask for a number "between five and twelve" you keep getting seven.

And this was the result:

answer 5 --> 1 people.
answer 6 --> 1 people.
answer 7 --> 3 people.
answer 8 --> 2 people.
answer 9 --> 4 people.
answer 10 --> 1 people.
answer 11 --> 0 people.
answer 12 --> 0 people.

Or, more graphically,

Group C:


Funny, isn't it? Seven is not any more the preferred number! It would be interesting to check this with a larger group of people, to make sure that fluctuations are not fooling us.

Conclusion: Recognising that groups of 12 people to choose between eight numbers is not a great deal, we can provisionally say that

• asking for subtractions before making the "important" question is not the key point for the number the people has to pick up when asked for a number between twelve and five (though there can be some influence smaller that the error of this experiment)
  
  
• it seems that it is more important here the order we ask for the numbers.
  

Maybe this is because are used to subtract when we have a larger number preceding a smaller number. Of course, to check this idea a bit better... an experiment with different numbers (other than 12 and 5) is needed.

Of course!