Solution to Puzzle #180: White and Black Beans

This was one of the simpler puzzles and I got correct answers from many. This included Mahi Saraf, Anahat (Prakhar’s son), Vishal Poddar, Pratik Poddar and Praneeth. Well done all.

The answer is “white”. I am borrowing the answer from Mahi.

The bean remaining will always be white because every time we are doing something to the pot we are taking out only 0 or even number of white beans. As we started with an odd number of white beans, the bean that will be left will be white.

Hope you all enjoyed the puzzle!

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Puzzle #181: Hidden Cross

This is a wonderful puzzle – one of those “2 minute” puzzles that will be fun to do as a family. I have taken this from Martin Gardner’s Colossal Book of Short Puzzles and Problems. It appeared originally in the 1978 issue of the magazine “Games”.

The task is to trace in the larger figure a shape geometrically similar to the smaller one shown besides it.

Graphic for Puzzle #181

Graphic for Puzzle #181

As always, please send your answers as comments within the blog (preferred), or send an e-mail to alokgoyal_2001@yahoo.com. Please do share the puzzle with others if you like, and please also send puzzles that you have come across that you think I can share in this blog.

Happy tracing!

 

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Solution to Puzzle #179: Crossing the River

This was a very simple puzzle, and therefore I guess I did not challenge enough minds. I got a correct answer from Mahi Saraf – well done Mahi! I would still encourage parents to show this puzzle to their children, it will be an interesting one for children.

I am taking the liberty of reproducing the answer from Mahi.

A=90kg, B=80kg, C=60kg, D=40kg, Supplies=20kg

Here is the sequence:
  1. D and C go
  2. C comes back
  3. A goes
  4. D comes back
  5. C and D go
  6. C comes back
  7. B and supplies go
  8. D comes back
  9. C and D go
Hope you all enjoyed the puzzle!
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Puzzle #180: White and Black Beans

This is a wonderful puzzle for all ages from Alok Mittal’s weekly Math Circles class. The puzzle is doable by everyone, so please do try yourself and also have your children try this one.

A pot contains 75 white beans and 150 black ones. Next to the pot is a large pile of black beans. A somewhat demented cook removes the beans from the pot, one at a time, according to the following strange rule: He removes two beans from the pot at random. If at least one of the beans is black, he places it on the bean-pile and drops the other bean, no matter what color, back in the pot. If both beans are white, on the other hand, he discards both of them and removes one black bean from the pile and drops it in the pot. At each turn of this procedure, the pot has one less bean in it. Eventually, just one bean is left in the pot. What color is it?

As always, please send your answers as comments within the blog (preferred), or send an e-mail to alokgoyal_2001@yahoo.com. Please do share the puzzle with others if you like, and please also send puzzles that you have come across that you think I can share in this blog.

Happy bean counting!

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Solution to Puzzle #178: Minarets of Samarra

This was a beautiful geometry arrangement puzzle, and I received correct answers from three people – Vishal Poddar, Mahi Saraf and Suman Saraf – well done all!

Am reproducing the answers from all of them, please notice the patters of 3 being repeated in all of them.

Solution to Puzzle #178

Solution to Puzzle #178

Hope you all enjoyed the puzzle!

 

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Puzzle #179: Crossing the River

After a rather difficult puzzle, here is an easy one, should take just a few minutes even for children, please do try! This one is from the site mathsisfun.com.

Four adventurers (Alex, Brook, Chris and Dusty) need to cross a river in a small canoe.

The canoe can only carry 100kg.

Alex weighs 90kg, Brook weighs 80kg, Chris weighs 60kg and Dusty weighs 40 kg, and they have 20kg of supplies.

How do they get across?

As always, please send your answers as comments within the blog (preferred), or send an e-mail to alokgoyal_2001@yahoo.com. Please do share the puzzle with others if you like, and please also send puzzles that you have come across that you think I can share in this blog.

Happy crossing!

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Solution to Puzzle #177: How Many Times Can You Measure

This was a difficult puzzle. I got the correct answer only from Suman Saraf, and another correct answer to a slightly modified puzzle from Pratik Poddar.

You can measure 23 different lengths. Here’s an explanation, taken from the site fivethrityeight.com.

The first trick is realizing that you can record 30 minutes by burning both ends of one rope. Since you know it takes an hour to burn through a rope from one end to the other, once the two burns meet you know each will have burned through 30 minutes of rope (even though they might not meet at the center of the rope thanks to the non-constant burn rates). You also have to consider burning the ends of the ropes at different times. For example, to measure 45 minutes, you can burn both ends of the first rope and one end of the second rope. After 30 minutes have passed, you can burn the other end of the second rope, making 45 minutes.

The calculation gets very tricky as we add ropes. Below are the possible lengths of time (in minutes, not including zero):

One rope: 30 and 60

Two ropes: 30, 45, 60, 90 and 120

Three ropes: 30, 45, 52.5, 60, 67.5, 75, 90, 105, 120, 150 and 180

Four ropes: 30, 45, 52.5, 56.5, 60, 67.5, 71.25, 75, 78.75, 82.5, 86.25, 90, 97.5, 105, 112.5, 120, 127.5, 135, 150, 165, 180, 210 and 240

Some of these time points are difficult to determine. For example, below are the many steps it takes to measure precisely 71.25 minutes with four ropes. (Let ri represent rope i.)

  1. Light both ends of r1, one end of r2, and one end of r3
  2. 30 minutes pass (r1 burned through)
  3. Light the other end of r2 and one end of r4
  4. 15 minutes pass (r2 burned through)
  5. Light the other end of r3
  6. 7.5 minutes pass (r3 burned through)
  7. Light the other end of r4
  8. 18.75 minutes pass (r4 burned through)

The total time passed is 71.25 minutes (=30+15+7.5+18.75).

From the site fivethirtyeight, I found a solver (Thomas Conroy) who wrote a program in C++ to find the lengths of time, which he was kind enough to post to GitHub.

Finally,  the following is the general solution for the number of lengths of time, T, you can measure with N ropes:

T = 3 * 2^(N-1) – 1

Hope you all enjoyed the puzzle!

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