This week's maths problem

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Problem of the week

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On Friday of each week we will post a new maths challenge.
You may submit your solution by the following Thursday.
Each problem contains two different levels of difficulty. You will be awarded from 2 to 12 points for a full answer.
Each series contains 12 problems before the stage winner is determined.
Your score will be published here.

In each series we will give away 3 books as prices. The prices will be drawn by lot among the best ten participants of the series. The books are kindly sponsored by Buchdienst Rattei of Chemnitz.

Suggestions for problems are welcome.

Deadline is 2nd of May 2019.

German version - Italian version - French version - Spanish version

Series 51

problem 603

“In an last year’s issue of the “Wurzel” mathematical magazine → http://wurzel.org/ ← I read something about a special kind of sorting problem. I have reconstructed this with some coloured cuboids.”
“As I see the cuboids are all of different size. I guess they have to be stacked from big to small.”
“That’s right. To do this I have a very thin piece of wood, that I can insert between any two cuboids an so turn around whatever is on top of this slat. When you look at the three depicted possibilities this means that that I don’t have to do a thing about the first stack. The two other stacks can each be turned in one go. I just have to insert the slat under the red cuboid.”
“So it’s allowed to turn a complete stack?” Bernd asked.
“Yes”, his sister answereed.

Here you can see five cuboids of different sizes (A>B>C>D>E). How many different ways are there to stack them? How many of these stacks can be turned into a correct stack in one go? - (2+2 blue points)
How many moves z do you need at most to sort a stack of n cuboids of different size? (Some assume z<=2n-3 for n>1, but is there proof?) - 4 red points.
Another red point is awarded for a stack of five cuboids (n=5) that can be sorted in exactly 5 moves.

Solving the picture-puzzle will get you two extra blue points, provided you also got points doing the regular maths problem. The rule for each picture puzzle is: Each icon represents one digit, same icons, same digits, different icons, different digits.  ©HRGauern[at]@t-online.de

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adress:
Thomas Jahre
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