15 Feb 2013

Prize Maths Quiz: A Tricky Tower (PMQ6)

Alice was feeling sick and lying in bed in the middle of the afternoon. She hated feeling sick and she hated lying in bed. Well, that wasn’t entirely true; she loved feeling soft and warm in her bed, but today she was feeling hot and irritated.

Bill suddenly bounced through the door.

“Hello Alice! How are you? I just popped round to cheer you up a bit – Mother said you had a slight fever.”

Bill was being chirpy; it seemed like bad manners to be chirpy on a day when Alice wasn’t.

“Look, I brought you a present!” Bill handed Alice a big box covered in coloured triangles and shapes. Before Alice had a chance to ask, Bill volunteered to explain, “It’s a jigsaw puzzle. A big jigsaw puzzle! I hope you like it.”

Alice thought Bill wanted a pat on the head and a fresh bone. The pattern on the front looked interesting but, as Alice opened the box, she frowned.



“What are all these squiggly bits and pieces?” Before Bill had the chance to state the obvious, “I know, it’s a jigsaw! But they’ve ruined the pattern!” Alice read the side of the box and slowly pronounced, “Pen-rose ti-ling.”

“What’s a Penrose?”

“I don’t know.” said Alice, in a weary voice. She hated jigsaw puzzles anyway. What was the point of showing you what the picture was going to be? Where’s the fun in that? But this time she hated it because it had ruined the pretty pattern with lots of small ugly pieces. Alice looked at Bill, thinking that she should be kind to him; after all, he did come to cheer her up.

“OK, why don’t you sit down here next to me and help me do the puzzle?”

Bill’s face lit up like a fun fair. As he busied himself sorting out the pieces, Alice stretched herself out, trying to find a comfy position. Within a few short minutes she was in deep sleep.


The Background

This puzzle is based upon the Tower of Hanoi (or Tower of Brahma) game. If you have never played this before, then now is the time to become familiar with how to solve the classic Tower of Hanoi.

The Tricky Tower is a little bit harder.


 

The Question

The Tricky Tower above has very similar rules. The Tower has six discs that are coloured as shown. The aim of the game is to move the entire tower from the left-hand column onto the right-hand column in the fewest number of moves. Each move consists in moving one disc from the top of a column and placing it on top of another column. No disc can be placed on top of a smaller disc, and no disc can be placed on top of a disc of the same colour.

The question is: what is the minimum number of moves necessary to move this 6-disc coloured Tricky Tower from the left to the right column?

Also, what is the layout of the game after move number 50?

In order to write the sequence of moves quickly, let the starting position be (6,0,0). This means 6 discs are on the left column and none in the middle and right columns. The first few moves could be written as:
1.    (510)
2.    (411)
3.    (402) and so on.

All you need to do is show how many discs are on each column after the 50th move has been completed.

Just to repeat, you need to answer both questions:

a)    What is the minimum number of moves necessary to move the whole tower from the left to the right column given the rules above, and

b)    What is the layout of the discs after move number 50?

Just as in the standard Tower of Hanoi, it may help you to try this with 4 and 5 discs before attempting the full 6 discs. There are also two different starting positions for the 5-disc tower: does it make a difference?


Have fun!




How to Enter

Send your complete solution by email to [competition now closed. email removed. feel free to discuss the question in the comments below.] This email address shall be removed after the competition closes to avoid spam. This PMQ6 competition closes on Sunday 17 February at 23:59 GMT.

The Prize

The prizes for this PMQ6 are 3 free places in our Online Classroom for 2 months. The very first correct solution will receive a prize plus two others randomly selected from all the other correct answers. The email time stamp shall determine the order of entries received.
 
Quick Rules

Look at the expanded rules on our PMQ page.
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