I have a problem, if the number of the figures of the numbers in the hands is finite, then the function f must be 0 to the minimum value in the hands and 1 to the maximum value of the number in the hands, so that there is not a tails of the distribution that don’t cover the random number (for example z can be generated in a region where f^-1(x) have not random numbers in the hands, with the limit case of a single figure); if the number of the figures is infinite, then the reading time is infinite, and there is not solution in a finite time, so that I think that there is not a real solution. ]]>

I enjoyed Poundstone’s book *Labyrinths of Reason*, and may have looked at *Fortune’s Formula* in a book store, but not *Are You Smart Enough to Work at Google?*. I should try it.

I have a student, by the way, who is smart enough to work at Google. I’m reading and correcting his PhD thesis now, even though I’m not smart enough—or something enough—to work at Google myself.

I suspect the reason you find the puzzle here tough is that it deals with a situation that doesn’t come up much in practice. Furthermore, it’s one in which the goal is to *win more than 50% of the time no matter what, but possibly by a microscopic amount more than 50%*.

In real-life situations one is more interested in doing better by a significant amount in average-case situations, or perhaps doing better by a significant amount in every situation, not doing better in every situation but perhaps by a microscopic amount.

]]>I read not so much for the puzzles, (turns out I am not smart enough to work at google. I needed to read the answers.) but rather to understand how smart companies hire quality employees (Don’t hire people who waste time on obscure math blogs when they should be mopping the floor). It also explains, buy the way, why we ‘google’ something, not ‘googol’ it. As it happens you can be a brilliant business person and a bad speller. Sadly the converse is not true.

Here is my favorite puzzle from the book: You have 367 students in your pre-Calc section take a quiz. They are scored from 0 – 100. Write the psuedo-code for an algorithm to sort the papers by score. Hint: your mother will do a better job than you. (If you think I got the wording wrong, and wording is everything in puzzles, you may be right. Read the book to find out!)

What I really wanted to ask here in this forum is this: why I am so bad at this puzzle, but seemingly so good at my job? I am an engineer. I spend lots of time using math. In fact, sitting on my desk next to me is “Statistics for Experimenters” (book recommendation #2). I use this about 2-3 times a year. I think I understand it. I design about 10 experiments a year, run them, analyze the results. Based on those results, change what I do in my factory. People pay me money to use statistics and probability. So why, when I am confronted with this puzzle is it so completely alien to me? This is something that baffles me. Is it the case that there are opportunities to think like this all the time in my job, but I am too dumb to even formulate the problem in a way that would allow me to do so? Or is it the case that puzzles like this are just few and far between in the real world.

Last week I saw a truck stuck under a bridge, jammed tight, a rental of course! I wanted to shout, hey, do you want a good puzzle? In stead I just gave them the answer (do you know what it is?) Those puzzles come up all the time for me. But never something like the one this post is about. Any thoughts as to why? Anyone?

Final book recommendation: “Fortune’s Formula”, also by Poundstone. A great read about some amazing mathematicians, including Shannon. These guys worked at AT&T when it was the cool-Google-place-to-be, and they sure seemed to use cleverness all the time to solve problems.

]]>Don’t we have to marginalize over all possible choices of ?

No, we pick any specific obeying the three rules listed, and prove that this gives a way of winning with probability more than 50%, no matter what numbers the first player has chosen.

Alternatively, if we fix don’t we have to take into account the probability that the numbers chosen are “very large” or “very small?”

The proof takes this into account.

How do we know that such considerations won’t force the probability back down to ?

It can be driven arbitrarily close to , but it will necessarily exceed

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