Solution to Puzzle #65: The On-Off Game

Posted on July 6, 2014 by Alok Goyal

Welcome back…I was traveling for a month and hence no posts…but normal service resumes!

I got many answers to this puzzle, and most of them correct. While I had done this myself, I got a nice and easy explanation for this puzzle from Alok Mittal first from his Mathematical Cricles class. Aman Singla wrote a program to solve this (cheating!). Many others did it correctly that includes Radhika Goyal, Deepak Goyal, Shruti Mittal, Amritansh Raghav and Shivam Mittal. Well done everyone and thanks for sending your answers!

The answer is – All square numbered bulbs will be on, rest will be off. The reason is that only squares have an odd number of factors and therefore they get toggled an odd number of times. Here is a link to the video answer for children:

Hope you all enjoyed the puzzle!

 

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Puzzle #65: The On-Off Game

Posted on May 31, 2014 by Alok Goyal

This is a relatively common puzzle and available in many puzzle books. Interesting one, good to try with kids.

There are 100 light switches, all of them are off. First, you walk by them, turning all of them on. Next, you walk by them turning every other one off. Then, you walk by them changing every third one. On your 4th pass, you change every 4th one.

You repeat this for 100 passes. At the end, how many lights will be on?

As always, please send your answers directly to me at alokgoyal_2001@yahoo.com. If you like the puzzle, please share it with others. If you have interesting puzzles to share, please send them to me at my e-mail given above.

Happy switching!

 

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Solution to Puzzle #64: Unique Country

Posted on May 31, 2014 by Alok Goyal

I think I went a little bit overboard with the puzzle I posted last week…did not get any correct answer. Was hoping that there must be some Computer Scientists who will send me the answer!

Anyway the answer is that there cannot be 100 roads in a country where every city has exactly 3 roads connecting the city. The solution lies in two concepts – simple graph theory, and divisibility by 3.

Think of this country as a graph, where each road is an edge, and each city is a vertex. We therefore need to build a graph with 100 edges where the degree of each vertex is exactly 3…the keyword being “exactly”. If we were to add the degrees of each of the vertex (or city), then the sum should be 200, as each edge (or road) contributes to 2 degrees (one each for the two cities it connects). Since 200 is not divisible by three, such a graph is not possible.

Hope you enjoyed the puzzle!

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Puzzle #64: Unique Country

Posted on May 24, 2014 by Alok Goyal

This is a very nice application of graph theory for children, a puzzle I picked up from the book “Mathematical Circles”, by Dmitri Fomin, Sergey Genkin and Ilia Itenberg.

There is a unique country (lets call it Strangistan), where there are exactly 100 roads between pair of cities and each city has exactly 3 roads going out of it. Show how that is possible or show that it is not possible.

I will be traveling on work and vacation from next weekend, and hence will not be very regular in my posts until the end of June.

As always, please send your answers directly to me at alokgoyal_2001@yahoo.com. If you like the puzzle, please share it with others. If you have interesting puzzles to share, please send them to me at my e-mail given above.

Happy traveling on roads in Strangistan.

 

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Solution to Puzzle #63: The Funny Equation

Posted on May 24, 2014 by Alok Goyal

Thanks for a great response…got many correct answers, including some first timers. Anirudh Baddepudi was the first one to come back with the correct answer. Also had correct answers from Adila Sebastian, Rajesh Thakur, Anshu Goyal and Anubhav Gupta.

Correct answer was the following:

9 – 5 = 4

6 / 3 = 2

7 + 1 = 8

To explain the answer to children, here is a video for the solution:

Hope you enjoyed the puzzle!

 

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Puzzle #63: The Funny Equation

Posted on May 17, 2014 by Alok Goyal

Great small numbers puzzle for children! I found this one in a new book that I bought last week “World’s Toughest Puzzles” by Charles Barry Townsend. Puzzles are not as tough as the name of the book suggests 🙂

One is normally used to seeing one dimensional equations. However, here is a 2-dimensional equation:

Puzzle 63 - question

Puzzle 63 – question

As you can see above, the equation(s) above do not match. Your task is to change the locations of numbers 1..9 in such a way that all the equations match!

As always, please send your answers directly to me at alokgoyal_2001@yahoo.com. If you like the puzzle, please share it with others. If you have interesting puzzles to share, please send them to me at my e-mail given above.

Happy equating!

 

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Solution to Puzzle #62: How Much Did I Drink?

Posted on May 17, 2014 by Alok Goyal

Many people did this one correctly. First one to send the right answer was Ashish Gupta, my colleague at Helion. Also got correct answer from Pooja Goyal and Anirudh Baddepudi.

Many people typically attempt this problem by doing a series, i.e. first person drinks 1/2, second one drinks 1/4, then the first one drinks 1/8 again, etc. While that can help you come at a series and one can use algebra to arrive at the right answer, the simpler way to think about the problem is the following – in each set of one chance each by the two people, “I” drinks twice of “You” (1/2 and 1/4 respectively in the first chance, and 1/8 and 1/16 respectively in the second chance, etc.), and therefore that will hold true for the overall pint of juice. “I” will therefore drink 2/3 and “You” will drink 1/3.

Ashish also solved it without resorting to Algebra where he put the two series on top of each other and realized that one is twice of the other, and arriving at the same answer.

Hope you enjoyed the puzzle!

 

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Puzzle #62: How Much Did I Drink?

Posted on May 10, 2014 by Alok Goyal

This is another puzzle I picked up from Brilliant.org posted by Can Hou and the puzzle is originally from Oxford University interviews.

I have a pint of orange juice. I drink half of it and then give it to you. You drink half of what’s remaining, and then pass it back to me. I drink half of what’s remaining and then give it to you. This process continues forever. How many pints did I drink in total?

As always, please send your answers directly to me at alokgoyal_2001@yahoo.com. If you like the puzzle, please share it with others. If you have interesting puzzles to share, please send them to me at my e-mail given above.

Happy drinking (only water)!

 

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Solution to Puzzle #61: How Many Pearls

Posted on May 10, 2014 by Alok Goyal

Many of you came back with the correct answer this time – thanks. I got the first correct from Shreya Mantripragada (9 year old, Bay Area). Also got correct answer from Anubhav Garg who explained his answer really well, Shruti Mittal and Anisha Sharma Goyal, though it took some pushing from my side!

The correct answer is 101 as many of you pointed out. Here is the logic:

From rule #1 (number -1 is a perfect square) and #3 (number -1 is divisible by 5) one can conclude that the number must be divisible by 25.

From rule #4, we know that number-5 is divisible by 4 (since it is divisible by 8) and therefore number -1 will also be divisible by 4.

From the above two conclusions, we can conclude that number -1 needs to be divisible by 25 as well as 4 (both of which are squares as well), therefore the smallest such ( number -1) can be 100, and therefore the smallest number can be 101. Quick inspection will indicate that all the conditions are satisfied and therefore 101 is the smallest such answer.

Hope you all enjoyed the puzzle!

 

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Puzzle #61: How Many Pearls

Posted on May 4, 2014 by Alok Goyal

This is a simple puzzle, and would love for children to try it! I picked this up from a new website recommended to me by my wife – Brilliant.org…has a great selection of puzzles. This specific puzzle was posted by Victor Paes Plinio from Brazil. Here is the puzzle:

A king has a chest that is filled with pearls. He wants to find a good and reliable treasurer to take care of his pearls. As such, he created a riddle for his villagers to solve; whosever solved it correctly first will become the new royal treasurer. The riddle is as follows:

“In this chest I have a number of pearls.
If I remove one, the number becomes a perfect square.
If I remove 2, the number becomes a multiple by 9.
If I divide these pearls amongst 5 knights equally, there is 1 left over.
If I divide them amongst 8 knights, then there are 5 left over.
The number of pearls is prime.”

What is the minimum number of pearls in the king’s chest?

As always, please send your answers directly to me at alokgoyal_2001@yahoo.com. If you like the puzzle, please share it with others. If you have interesting puzzles to share, please send them to me at my e-mail given above.

Happy solving!

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