Gary posed a math question in comments.
Since we are currently discussing interesting shortcuts to solve problems, here is one for you math enthusiasts:
How can one easily compute the sum of all of the numbers from X to Y in literally seconds (versus adding them all up sequentially)? For instance, what is the sum of all the numbers between 1 and 10. Or between 1 and 20? It takes no longer than a few seconds to find the answer.
Here’s another problem : I play the Queen of Hearts at the Elks Lodge. If they pick your ticket which has a number from 1 to 52 on the first drawing, and the number on your ticket corresponds to the number associated with the Queen of Hearts you win the jackpot. Tickets cost $2.00. Of course at the first drawing there are 52 cards in the deck , so someone’s chance of winning is 1/52. The next drawing assuming someone didn’t win at the first drawing is 1/51, etc. I want to know before the first drawing, how many cards must be drawn before the odds are 50/50. The answer is n where ∑1/52 + 1/51 +…1/n = 1/2. Of course the odds before any particular drawing is 1/n where n is the number of cards left in the deck. The Elks cap the jackpot at $15,000 because the money (so they tell me) goes to any of their several charities. Three months ago they drew my ticket and my card was the JACK OF HEARTS! Geeze! The Piranha Club should do the Queen of Hearts.
On the island, Anna Maria Island where I grew up, at the Moose Lodge the jackpot is over $250,000! That’s because it’s the biggest membership in all the Mooses in the country. And it’s the biggest because it’s smack on the beach. When I was a very young man what is now the Moose Lodge was an Elks Lodge. I spent the best New Years Eve in my life there once.
I’ll be at the Elks this Wednesday. There are 4 cards left.
Makes my head hurt 🙂
Used to play this one at Rotary Club with a bag of marbles – 50 white ones and one black. Black one got the pot. Huge payouts!
The Queen of Hearts question reminds me of one that our 7th grade social studies teacher gave us as an example on population growth:
“I got a water lily to put in my backyard pond. The next day, there were two lilies in the pond, and the day after, there were four. According to my calculations, at this rate, on day 30, my pond will be completely full of lilies. I believe that aesthetically, the pond will be most pleasing when it is half-covered with lilies – On what day do I need clear lilies out of the pond to keep the level I want.” Not terribly complicated to figure out now, but to the 7th grade mind, quite the puzzler.
Another probability puzzle is the birthday paradox, which I first heard from Amarillo Slim, the famous/infamous poker player who bet on all sorts of other things:
“In a group of people, I bet that there are at least two people who have the same birthday – day and month, but not necessarily year How many people need to be in the group for the odds to be 50:50 that two people have the same birthday?”
Hint: The relevant calculation is (1 – p(nobody has the same birthday))
On a different tack, Slim used to put out a line of sugar cubes in the presence of houseflies, and bet on which cube a fly would land on first. How did he rig the game to know which cube was favored?
On the day the pond is first full you take out half the lilies. Then every day afterwards you take out half. Is this the answer? As for the birthday problem, that’s a classic. I’m too tired right now to work it out, but I seem to recall the answer is about 26. I graduated with a BA in math (Honors taboot) so I used to be smart. The most difficult course was a senior course in number theory. I thought I still had the textbook, but I can’t find it just now. The best part of it was my professor, Marion Tinsley. He was brilliant, but a lousy teacher. I was the only one in the class who knew what was going on and the only one who got an A. I want you to Google Marion Tinsley. He was and is still recognized as the greatest checker player of all time. He told me that once he played 26 people simultaneously while blindfolded and beat them all. He was kind of weird. He was a devout Baptist. He called me “Amazing Grace”
Finally looked up Tinsley. An amazing life story. I found this article about the computer program developed to try and beat him.
https://www.theatlantic.com/technology/archive/2017/07/marion-tinsley-checkers/534111/
Yes, you’ve got to clear the lilies on days 29 and 30 to keep the pond from getting choked. The standard 7th grade answer is day 15, because we weren’t getting the difference between arithmetic and geometric series.
The birthday paradox answer is 23, which really throws people who haven’t done the math themselves. The counter puzzle is, “How many people do you need, to 100% guarantee that at least two share a birthday?”
And the answer to the sugar cubes is, that Slim would surreptitiously lick a finger and “adjust” a cube. There’s nothing to smell from dry sugar, but once it’s wet, the flies are attracted.
Right I looked it up. That was after I spent half an hour trying to simplify the series. It can’t be done. You can get a good approximation by taking an integral. As for the sugar cubes I figured he dipped one of them in dog poop. Wrong.