Derren Brown (Magician) and his odds claim

Hi all,

Pretty sure I've calculated this correctly but just need someone to verify. On Derren Brown's recent stage show, Svengali he does a trick whereby people have to choose various coloured blocks which ultimately produces three 4 digit codes and he claims the odds of them picking the ones they do are 1 in 280,063,590,219 which immediately struck me as strange given the simplicity of the choices.

The trick and Derren's maths can be seen here...

http://www.youtube.com/watch?v=2RJY2-YvIYA (51 minutes onward, might not be view-able to people outside the UK)

If you can't see it, the game that he attributes those odds to is this....

Three people stand behind a table of 4 rows of coloured blocks. There is a row of 10 red blocks numbered 0-9, the same for yellow, blue and green blocks (so 40 blocks in total). The three people choose blocks at 'random' (well not really random if you see it but just go with it for the sake of working out the odds), choosing first 1 red block each, then 1 yellow block each followed by 1 green each and 1 blue block each. After which all three people have created a 4 digit number comprised of four differently coloured blocks.

Now, I work out the odds of the three picking three specific 'codes' as in 1 in 268,738,560,000. First all three have to pick a red block each. The number of possible combinations they could pick between them for red blocks is 720 (10x9x8). There are also 720 possible combinations of yellow blocks they could have taken, 720 green and 720 blue.

Therefore, the total possible combinations are (720x720 x720x720) = 268,738,560,000.

Am I correct or is Derren? (P.S given the codes they pick match his stated odds I suspect those aren't the odds and he's using that number purely for the trick).

I haven't worked out a number myself, but I think you are calculating permutations instead of combinations. Even if you had correct combinations for each of the four colors, I don't know right off the top that you can just multiply the four numbers together in this case -- I'd have to think about that for a few minutes. Maybe I will get my brain turned on (which is getting harder each day) and give this a try later. However, there are a lot of mathematicians around this forum, and I'm not one of them.

Unfortunately, the video is "not available for viewing in my country" so I haven't seen it. (I looked online, and couldn't find it.) If anyone can find it elsewhere, please post the URL.

However, based upon your description, I think I fully understand the game/trick. Question... does Derren not know what numbered blocks the people are picking and does he then "figure it out?"

If arrive at the same number as you, but differently.

The first person has a total of 10,000 different possible codes. (10 red choices x 10 yellow choices x 10 blue choices x 10 green choices)

The second player has one less block of each color to choose from. Thus his/her possible permutations is reduced to 9^4, or 6,561.

The third player has one less block of each color to choose from. Thus his/her possible permutations is reduced to 8^4, or 4,096.

So, the chances of predicting the first players code (1/10,000) AND predicting the second player's code (1/6,561) AND predicting the third player's code (1/4,096) is the product of all three... or 1 out of 268,738,560,000.

And yes, Doc is correct. If the order of the numbered blocks is important, and it sounds like it is, then we are talking about permutations, not combinations.

(We sometimes use the word "combination" too loosely. For example, it really should be called a permutation lock, not a combination lock.)

http://www.youtube.com/watch?v=8CkNXibZDYQ&feature=relmfu - Looks like it here, about 9 minutes in. He claims 280 Billion, not 208 Billion in that video.

Part 5 has the follow up : http://www.youtube.com/watch?v=SkBEr3X3L_k&feature=relmfu

I think he had to use a "fake number" so it fits the "trick".

A quick check on wiki suggests that the Targ experiment didn't happen until the 70's, for example. That said, I've seen Derren Brown live, and as a show man, who uses mentalism to cover for traditional tricks, and psychology as the distractor for sleight of hand, he's very enjoyable to watch. He's also honest, he will tell you he uses deception in his act, so just cos he says something... doesn't mean it's actually true... it's part of the trick.

Quote: thecesspit

I think he had to use a "fake number" so it fits the "trick".

That's what I said in the OP, I wanted to check others agree as I think I know how the trick is done but wanted some maths whizzes to confirm my thinking on the odds to confirm it.

Quote: thecesspit

A quick check on wiki suggests that the Targ experiment didn't happen until the 70's, for example.

Can you post a link? I've been trying to find out about this so-called 'Targ experiment' and couldn't find anything specific.

Quote: thecesspit

http://www.youtube.com/watch?v=8CkNXibZDYQ&feature=relmfu - Looks like it here, about 9 minutes in. He claims 280 Billion, not 208 Billion in that video.

Part 5 has the follow up : http://www.youtube.com/watch?v=SkBEr3X3L_k&feature=relmfu

My bad, typo. I've corrected the OP to 280 billion.

Quote: EdCollins

And yes, Doc is correct. If the order of the numbered blocks is important, and it sounds like it is, then we are talking about permutations, not combinations.

(We sometimes use the word "combination" too loosely. For example, it really should be called a permutation lock, not a combination lock.)

My apologies, whilst I'm quite good with numbers I'm no mathematician so don't always use the right terms as it were. When I used the word 'combination' I meant it in the same way you ask "what is the combination to that lock" whereby you are looking for a specific 4 digit number. So 1234 would be a different 'combination' to '4321' in that sense.

I didn't watch the video, but it the blocks are coloured and they can be rearranged to form a code (blue 8, red 7, yellow 3, green 4), then we are talking about permutations. If this is the case, both OP and magician are wrong, and there are even more possibilities.

But I got the impression that the blocks are not being rearranged, we are just choosing one block from each row, and they are coloured so they are pretty to look at. Then while the code itself is a permutation of numbers, it is also a combination of choices.

Now that the semantics are out of the way, the OPs math looks correct.

Quote: dwheatley

I didn't watch the video, but it the blocks are coloured and they can be rearranged to form a code (blue 8, red 7, yellow 3, green 4), then we are talking about permutations. If this is the case, both OP and magician are wrong, and there are even more possibilities.

But I got the impression that the blocks are not being rearranged, we are just choosing one block from each row, and they are coloured so they are pretty to look at. Then while the code itself is a permutation of numbers, it is also a combination of choices.

Now that the semantics are out of the way, the OPs math looks correct.

Sorry if I wasn't clear in my OP. Each person can NOT take more than one of each colour, so person 1 cannot pick four red blocks, he must first take 1 red block, then take 1 yellow block, then 1 green block and finally one blue block. All three people doing thus MUST end up with a 4 digit number comprised of 4 different coloured blocks. Also because there is only one of each block, no two people can end up with the same coloured number in their final 4 digit code (only one person can take the red number 3 block for example)

Add red block is always in the same position in the number.

http://en.wikipedia.org/wiki/Russell_Targ - wiki entry on Russell Targ, which talks about the Stanford Experiements in the 70's.

Quote: thecesspit

http://en.wikipedia.org/wiki/Russell_Targ - wiki entry on Russell Targ, which talks about the Stanford Experiements in the 70's.

Yeah I'd read that before but I was looking for information on the specific experiment Derren mentioned (transmission of a four digit number). I'm aware of Russell Targ and that he had a programme of various test in the 70s and 80s.