Puzzle #196: Bugs on a Board

Posted on September 17, 2017 by Alok Goyal

Yet another very fine puzzle from A Moscow Math Circle: Week by Week Problems by Sergey Dorichenko.

Each square of a 9 x 9 board has a bug sitting on it. On a signal, each bug crawls onto one of the squares which shares a side with the one the bug was on. (a) Prove that one of the squares is now empty. (b) Can the bugs move so that there would be exactly one empty square?

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 bugging!

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Solution to Puzzle #194: Adjacent Cells on a Chessboard

Posted on September 17, 2017 by Alok Goyal

I got only two correct answers for this puzzle – from Suman Saraf and Pratik Poddar – thank you both and well done!

I am taking the liberty of copying the answer from Pratik – I find it to be very succinctly articulated.

From square marked 1 to square marked 64, the maximum manhattan distance possible between the two squares is 14 and the maximum difference between any two numbers is 63. Since 63/14>4, there has to be at least one step where the jump is greater than 4. Hence proved.

Hope you all enjoyed the puzzle!

 

 

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Puzzle #195: 1989!

Posted on September 10, 2017 by Alok Goyal

This is a beautiful puzzle from Mathematical Circles (Russian Experience) by Dmitri Fomin, Sergey Genkin and Ilia Itenberg.

There numbers 1, 2, 3, ….. , 1989 are written on a blackboard. It is permitted to erase any two of them and replace them with their difference. Can this operation be used to obtain a situation where all the numbers on the blackboard are zeros?

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 zeroing in on the solution!

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Solution to Puzzle #193: Smallville Telephones

Posted on September 10, 2017 by Alok Goyal

This was a simple puzzle, and I got correct answers from many. These include Vishal Poddar, Pratik Poddar, Suman Saraf, Mahi Saraf and Anand Singhi. Well done all.

The answer is “Not Possible”. This is a simple application of graph theory. I am copying the answer directly from Vishal Poddar:

Lets consider 5 slots in each telephone. Every wire will occupy 2 slots.

Total slots 15*5=75.

Total wires require = 75/2 = 37.5 which is not an integer. Therefore not possible.

Hope you all enjoyed the puzzle!

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Puzzle #194: Adjacent Cells On a Chessboard

Posted on September 2, 2017 by Alok Goyal

This is another beauty of a puzzle from A Moscow Math Circle: Week by Week Problems by Sergey Dorichenko.

The numbers from 1 to 64 have been placed in each square on a chessboard such that each number appears exactly once. Prove that there will be two squares sharing an edge whose numbers will differ by more than 4.

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 Eid!

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Solution to Puzzle #192: Too Many 6’s

Posted on September 2, 2017 by Alok Goyal

This was a difficult manipulation of numbers, and I received 4 answers from Mahi Saraf, Anand Singhi, Pratik Poddar and Prakhar Prakash. Only two of these I found to be completely correct – from Mahi Saraf and Anad Singhi. That said, I found the approach from all of them to be very good and similar. Well done Mahi and Anand!

Here is the answer from Mahi, and I am also reproducing the answer from the original source.

6 + 66 + 666 …….
6 (1+11+111+1111…)
6(10-1/9 +100-1/9 + 1000-1/9 …..)
6/9 (10+100+1000……-100)
2/3(1(100 times) – 100)
2/3(1(98 times) followed by 010)
2({1(98 times) followed by 010} / 3)
2(037(32 times) followed by 03670)
 074(32 times) followed by 07340
Hope you all enjoyed the puzzle!
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Puzzle #193: Smallville Telephones

Posted on August 27, 2017 by Alok Goyal

This is a relatively simple puzzle from Mathematical Circles by Dmitri Fomin, Sergey Genkin and Ilia Itenberg. Please do share this with your children, they should enjoy this.

In Smallville, there are 15 telephones. Can they be connected by wires so that each telephone is connected with exactly 5 others?

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 calling!

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Solution to Puzzle #191: 1×4 Rectangular Stone Plates

Posted on August 27, 2017 by Alok Goyal

This was one of my favorite puzzles, though slightly difficult. Unfortunately I got only one correct answer this time, from Suman Saraf. Well done Suman!

The answer is “no”.

Solution to Puzzle #191

Look at the unusual coloring of the 10×10 board above. Please note that 52 of these are greyed out, while 48 are white. No matter how you place a 1×4 rectangular stone plate, it will always cover 2 grey cells and 2 white cells. Therefore if there was a way to cover the entire grid with 1×4 rectangular plate, one would need to have the same number of white and grey cells. But with our coloring, that is not the case.

Hope you all enjoyed the solution.

 

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Puzzle #192: Too Many 6’s

Posted on August 20, 2017 by Alok Goyal

This is another beautiful puzzle from A Moscow Math Circle, Week by Week Problem Sets by Sergey Dorichenko. The puzzle will require some manipulation of numbers, but is very solvable and you will be happy that you did it once you get there!

Find the sum

6 + 66 + 666 + 6666 + 66666 + …. + 66…66, if the last string has 100 6’s in 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 sixers!

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Solution to Puzzle #190: Escalator Race

Posted on August 20, 2017 by Alok Goyal

This was a tricky puzzle – I received two correct answers, one from Prakhar and another one from Suman Saraf. Thank you both!

Answer is that they will reach at the same time, provided certain conditions are met, see the detailed answer below.

Draw the escalators as semi-circles below, and indicate on each semicircle the direction of motion of the escalator.

Solution to Puzzle #190

Solution to Puzzle #190

When one of the girls steps off her escalator, she immediately steps on the other one. Thus we can assume that they are running on a moving circle. At the first, the girls are at one point of this circle and the hat is diametrically opposite. After that the girls start running towards the hat in opposite directions. Since their speeds relative to the escalators are equal, they will reach the hat simultaneously, unless the hat gets to the end of the escalator first. If this happens, the hat will leave the escalator altogether, and therefore Arushi, who started running down will get to it first.

Hope you all enjoyed the puzzle!

 

 

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