When I first saw this one, I thought it might be a chance to use some of the Erlang queueing equations, which I’ve known about for ages but never used seriously. Instead, I think it’s a bit easier than that. I missed the cutoff for having my name picked though!
My strategy of starting with the easy cases and building out to more complex cases was used successfully by other solvers. My analysis of the easy cases was correct, but there was a mistake in the way I built out. My diagram probably didn’t help; the one drawn by the winner didn’t distinguish between the larger and smaller piles, and I think that’s where my error crept in.
For example, when I considered (9, 5), I thought it was a losing position because I didn’t see that taking six coins from the larger pile produced (5, 3), since the larger pile became the smaller pile in that move. So the “simplification” I introduced, of specifying one pile as larger, while not wrong, made it easier for me to make a mistake. It helped to break my mental model by suggesting an identity for the piles (“larger” or “smaller”) that isn’t persistent, but depends on the progress of the game. Will try not to do that again!
Initially I thought this week’s Classic puzzle might be too tough to solve without some serious insights, but the more I thought about it, the more it seemed like I might be able to break it down to manageable pieces.
Most of the Classics look too difficult for me to be able to solve, but this week’s looked like I could approach it. No code required, either.
Here’s the question:
Five friends … are playing the … Lottery, in which each must choose exactly five numbers from 1 to 70. After they all picked their numbers, the first friend notices that no number was selected by two or more friends. Unimpressed, the second friend observes that all 25 selected numbers are composite (i.e., not prime). Not to be outdone, the third friend points out that each selected number has at least two distinct prime factors. After some more thinking, the fourth friend excitedly remarks that the product of selected numbers on each ticket is exactly the same. …
What is the product of the selected numbers on each ticket?
There might be a neat, elegant way of solving this, but I chipped away at it bit by bit.
Thought I’d see if I can get back into the groove of doing these, at least sometimes. This week‘s is geometry. I admit I got a bit of fear when I looked at it; geometry is something I haven’t used much since school, so it doesn’t come back so easily.
This week’s seems a little more involved than the Expresses usually are (although I’ll admit July 27th‘s really stumped me!):
Take a standard deck of cards, and pull out the numbered cards from one suit (the cards 2 through 10). Shuffle them, and then lay them face down in a row. Flip over the first card. Now guess whether the next card in the row is bigger or smaller. If you’re right, keep going.
If you play this game optimally, what’s the probability that you can get to the end without making any mistakes?
I got nowhere when I tried visualising this as a decision tree. Too wide and deep for me to understand it. Then I did the sensible thing, and broke it down into simpler problems. I also tried staying away from Excel for a while, a new one for me!
Say you have an “L” shape formed by two rectangles touching each other. These two rectangles could have any dimensions and they don’t have to be equal to each other in any way. (A few examples are shown below.)
Using only a straightedge and a pencil (no rulers, protractors or compasses), how can you draw a single straight line that cuts the L into two halves of exactly equal area, no matter what the dimensions of the L are? You can draw as many lines as you want to get to the solution, but the bisector itself can only be one single straight line.
This year’s World Cup has been chock full of exciting penalty shootouts. Historically, about 75 percent of soccer penalty kicks are successful. Given that number, what are the chances that a shootout goes past its fifth kick for each team and into the even more exciting sudden-death portion of the penalty-kick period?
I see two ways of solving this one. The first is to calculate it directly, by summing the probability of each possible draw. The second is to generate every possible outcome, and find the draws among them.