In prior Carnivals of Mathematics it has been a tradition to include trivia about the number the Carnival happens to be at in some way. With a desire to do something different for Math Teachers at Play, I offer a riddle. Younger solvers may have an advantage over older ones here.
Here is a pair of dice I own:
If I roll the dice and read the numbers off the top, is it more likely the numbers add up to be 2, or add up to be 12? Or are the two sums equally likely? Why?
Be careful before you answer!
Elementary Concepts and Arithmetic
Kendra has come up with a new game for teaching the reading of clocks. (The RummiKub variant is intriguing.)
The blog yofx gives a vacation photo of an unusual example of negative numbers in real life.
Many look at multiplication tables and see drudgery; Dan MacKinnon sees rainbows and hyperbolic arcs.
Speaking of multiplication, the folks at 360 have finished their series on 25 ways to multiply: 1-6, 7-12, 13-14, 15-17, 18, and 19-25.
Advanced Math
When you go to the pediatrician, do you notice mathematical error-correcting codes? Mark Dominus does.
Pat Ballew notes how one quadratic in particular can serve as a “graphic catalog” for all possible quadratics.
Dave Richeson finds complexity in even the simplest geometry and shares three cool facts about rotations of the circle.
Foxmaths 2.0 performs an clever bit of calculus on the function 
About Teaching Math
Want to keep your kids thinking about math over the summer? Kate Nowak is full of ideas at Math around the House.
Maria Miller reviews an online math practice system called Mathletics.
Tom DeRosa wants a TV show that changes the way we think about math.
Colleen King wonders if learning programming should be mandatory in education, and gives concrete suggestions for all the different grade levels.
Miscellaneous
Want to post math equations on Blogger/Blogspot like you’ve seen on WordPress? WatchMath has a solution.
Speaking of typesetting equations, John Cook owes Microsoft Word an apology.
This post about struggling with dyscalculia is a worthwhile read.
Finally, this is a series that appeared back in 2008, but it was new to me, so I hope it is new to some of you! Ron Doerfler wrote a three-part series on “lightning calculators”, people who can do astounding mathematical calculations in their head: The Players, The Methods, The Media. Especially fun (in part 3) is the deconstruction of a Daniel Tammet documentary and its “creative” editing.
Filed under: Education, Mathematics, Puzzles |
The Number Warrior 
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Oh neat! A lot of good stuff. I wonder who submitted me.
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I’m stumped on the riddle. Time for a hint?
I want the answer to be equally likely, but you warned me it was not so simple.
Here’s what I did: The probability that 1 and 1 show up is 1/36. The probability that 6 and 6 show up is 1/36. These are the only ways to get sums of 2 and of 12, and they are equally likely.
Clearly, I’ve missed something.
Jonathan
The picture is important.
Are you saying the roll of “1” and the roll of “2” together make 12?
No.
Perhaps because you’ve already rolled a 12, the possibility of rolling a 2 next is more likely.
Alas, also no.
I’d already figured the picture was important, but I can’t discern anything significant from it. The dice appear to be normal, fair, and equal, to the extent that one can tell from a blurry photo.
We can’t see the 4 and 1 sides, so perhaps there’s some trick to the dice involving that. Like, maybe there’s no 1 opposite the 6, so a roll of 2 is impossible. But that’d be cheesy, because there’s no real clue of any such thing.
It’s not anything you mention in your second paragraph.
For the riddle: is it that we know that there is a six on each die, but we don’t know whether or not there is a one on each die (because that side is not pictured, and conceivably they could be non-standard dice), so the probability of getting 12 is higher?
I need to learn to read not just the first comments, but also the recent ones.
The answer is that you are more likely to get “2”. This is because if you roll a 1 and a 1, you add up to get “2”. If you roll a 6, which is made up of two “1” shapes, you also get “2”.
I hope I am right.
Nope, there’s no trickery like that.
It’s hard to tell, but from the picture I think I notice a slight concavity in the material used to backfill the holes (12 of them for boxcars). So it is conceivable that there is overall less mass on the side with a ‘6’ than there is on the side with a ‘1’, which would lend a small bias in favor of the dice landing with the 1’s on the bottom and 6’s on top.
Yes.
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Surely the difference is so minor as to be far overshadowed by other factors (such as other minuscule irregularities in the dice). This seems rather like the specious claim that drains empty in different directions south of the equator vs north, owing to the Coriolis effect.
Have you actually done an experiment to show that there’s any real statistical difference?
Yes, I have witnessed the effect for myself. It is also well known in the gambling literature. (That’s where I first came across it — I think it may have been Scarne on Dice.)
The weight effect is generally far greater than any manufacture error in the dice (relatively speaking, as I said in the other post, you need at least 500 rolls before you have a chance of spotting anything), although it is a good point that cheap enough quality might override everything.
[…] Math Teachers at Play #12 is up and running, with a pleasing variety of posts to browse. […]
Huh, hadn’t heard of these before, but I’ve got a submission at:
http://www.discreteideas.com/2009/08/making-more-math-geeks/
Actually, there is the same exact odds of rolling 2 or 12 each and everytime the dice are thrown. The odds of rolling each are 1:36. This is why at a casino, the payout when you bet on either is 1:30; Sounds like great odds, but you are still the underdog by the odds of 1:6. Which means statistically, the casino has the better odds of winning by 6:5 overall, or for each roll. or 36:30.
Sir Chadwick, you make me wonder whether you even read the comments above. The dice pictured above are cheaply produced dice, not high-quality casino dice. With cheaply produced dice, holes are drilled and not too carefully backfilled, and this definitely introduces a bias. Statisticians familiar with both kinds of dice (such as Mosteller) are quite familiar with the phenomenon.
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