There are 5000 songs on your device, and you put it on random shuffle. Unlike some popular devices this player shuffles with replacement so that is there is a 1/5000 chance you hear the same song twice in a row. How many songs do you have to listen to guarantee a 99% chance you have heard all of the songs?
I came across the above problem while tutoring a proprietary derivative trader for his firm’s gaming class. After playing around with it, I realized that I wasn’t going to be able to solve it myself, at least in any reasonable time frame. So over the next few days, I sent it to half a dozen smart friends who enjoy problem-solving—all of them with at least one Ivy League degree, half of them engineers as undergrads.
Each of them brought something to the problem. But none of them brought everything.
Suggesting that reading is a hard part of problem-solving, two of my friends solved different problems than the one specified. One found the 50% probability of playing 100% of songs, using harmonic numbers without even realizing it. His was an elegant solution—a couple lines in Excel, no fancy math. While his approach didn’t give me the right answer, it did give me an intuitive feel for the problem’s mechanics and a way to check other answers since any answer that could get to the 99% probability of 100% of songs would also be able to return the 50% probability of 100% of songs, which he nailed.
The other friend built an inelegant but functional Excel model to randomly draw songs until reaching a 99% probability that 99% of songs had been played. Again, not the answer to the problem asked, but he gave me a working model that could be repurposed. More importantly, he pointed out what I was finding, that solving theoretically “is exhausting, and you run out at some point, no matter how smart you are.”
Which is exactly what those of us attempting to solve theoretically were finding. You can’t raise a number to the power of 5000 in Excel. Even Excel runs out. But one of my theoretical-solving friends gave a particularly clear explanation of how the problem worked with a smaller number of songs. The type of explanation that a reasonably smart person who hadn’t studied math since high school would be able to understand.
Finally, my sister’s boyfriend—a statistics graduate student—solved for the right answer to the right problem. When I opened his answer, I knew it was right even though he worked in R, a language I’ve never used, and explained the problem as if he were talking to another stats grad student. Thanks to my friends’ help, I had by that point a strong theoretical understanding of the problem and a way to check any answer.
Not to be outdone by my sister’s boyfriend, my dad provided the icing on the cake, writing a program to visualize the problem. He laid out 5000 cells and every time a song was picked, its cell changed colors. His method was slow, taking overnight to run 100 trials, but it allowed anyone to watch the problem being solved, to become curious about it, to develop a feel for how it worked.
So while all of these friends are out the right hand tail of the intelligence distribution, none of them got everything but all of them got something.
This process is also how Harvard Business School case discussions are intended to work. The professor leads the class through a discussion, moving from facts to analysis to broader themes. Everyone is expected to contribute something but no one contributes everything, especially not the professor whose role is to shepherd discussion—shutting it down, amplifying it, finding fresh viewpoints, and pushing back.
Yet, challenging problems are too rarely approached this way in the working world. Much knowledge work is still assigned to single solvers and is structured as if one person can theoretically understand, empirically solve, and lucidly explain a problem. Managers and teammates tend to engage at the end, reviewing work and pushing on assumptions, rather than at the beginning, brainstorming framing and different approaches. In fact, many employers consider this a model that “leverages” the team’s–and especially the manager’s–time.
But for tough problems or problems subject to judgment, an hour of six different people’s time is worth more than six hours of one person’s time. And broad team input is more valuable at the beginning of the project than at the end—appropriately facilitated, six cooks are more valuable than six tasters. So the next time your team needs to build a model, why not have everyone spend one hour sketching out how they would approach it before one person sits down to build it.
Because while at some point, we all run out, you don’t have to solve to help solve. And the best time to start helping is at the beginning.