I think there's an easier proof which says that a_6 is less then one. p_6 = a_6 * p_1 so p_6 < p_1. So P(12) < P(2). Then P(2) not= P(12) and hence such a dice can not be built. (not sure on the index of the a, might be 5) This proof generalizes to k-dices easily: p_k = p_1 * a_k. since a_k < 1, p_k < p_1. But then P(2*k) < P(1).
as we can see, Google back then, 10 years ago, was - just Google. Those were beautiful times when one could really Google something and get the result set related to search criteria. Maybe this could be the subject of Knuth's next book? "Why Google failed (like everything else)?"
Why does he have to use polynomials? If the number of faces on the dices are specified, like in this case, you can show that you get an overdetermined, coupled system of equations that doesn't have a solution. I didn't really watch the proof though, maybe it was a nice one... ^_^
Heads up, Knuth makes an error in writing out the dice math: p4 = 1/2 * 3/4 * 5/6 = 15/48 or 5/16, not 15/16. In other words he gets the unsimplified form and the simplified forms mixed up. Don Knuth really has forgotten more than I'll ever know.
Interesting, but somewhat confusing at times when he stutters too much and he skips or fast-forwards a few steps in the "proofs". I have no trouble with the math he is trying to explain but the way he does it isn't very easy to follow.
Here is a very good question for anybody who wants to come up with some good explanation: Why do so many very intelligent people, particularly mathematicians, stutter, some more so than others. In the last month or so, I have watched about a dozen or so brilliant mathematicians, and more than 50% of them stutter to some degree or another? Any good explanations would be welcome. Thanks.
you're asking for a direct correlation between a speech impediment and intelligence. here you go: www.csun.edu/~ainslab/readings/Anna/Barnard_1930_Relation%20of%20intelligence%20and%20personality%20to%20speech%20defects.pdf
Because of the way the brain works. Unlike other tissues like the muscle that can cope with increased functional load by increasing the size of the individual cells, or the liver - by increasing the number of its cells, the brain can do neither as it is confined within the walls of the skull, so the brain can't grow in size, at least not with substantial speed, it takes a while to remodel the bone of the skull. So the brain goes around that problem by reassigning the roles of its neurons - neurons performing tasks that have become obsolete (as those of the visual cortex of blind people), or less of importance, are assigned to task that are higher on the priority list - that is how the brain plasticity works. The second case is when the wiring of the brain during the its development, benefits some areas at the expense of others. The interplay between genetic make-up, environmental stimulation, and personal choice, will eventually determine which areas of the brain are going to be dominant over which ones and to what degree.
*_5.10 Tax Heaven 5.10 Financial Paradise 5.10 Free movement of people, goods, services and capital 5.10 State get out of economy 5.10 Multicurrency 5.10 Multilanguage 5.10 Right to keep and bear arms 5.10 Wealthy people 5.10 Libertarian idea 5.10 Balashov 5.10_*
And possibly the result of a life of thinking and pondering over speaking... (and possibly speaking about thinking). He also probably has thoughts and ideas intrude mid-speech and considers them more an instant, ergo, stuttering or odd speech patterns
Ahabite oh yeah what's definitely happening is he changes he mind about what he wants to say 20+ times a sentence and therefore can't get any words out. tons of people have gone through a phase of being like that until we learned how to deal with it and become better speakers. to become a better speaker you do not need to become a worse thinker. in fact, i'd say both skills lead to one another.
