Prize Maths Quiz: Nine Rooms Puzzle (PMQ29)

You are in a network of rooms laid out in a 3 by 3 grid (as shown in the diagram). Each room has a door connecting it to rooms adjacent to it. There are 12 doors in total and all are open at the start of the game. You are placed in one of the corner rooms and your aim is to visit each room once. When you leave a room, all the doors connected to that room are automatically closed and locked. You may finish your route at any convenient room.

In the middle of each room, there is a gold coin placed on a table. You must enter a room in order to pick up each coin. If you have successfully collected all nine gold coins, you will be magically teleported to freedom!

The diagram shows one possible winning route. Starting from the same room in the top left corner of the network, how many different routes are possible so as to visit each room once?

Now, that was far too easy! The real game is slightly more challenging. In the first room, next to the gold coin, are two dodecahedral dice, each numbered 1 to 12, and a small map of the network. Each door in the network has a number on it from 1 to 12 inclusive and the map shows the location of each numbered door. You roll the two dice until you have two distinct numbers; you may only roll again if the two numbers are the same. You will then hear the sound of two doors being locked closed; they are the doors that correspond to the two numbers on the dice.

What is the probability that you can still pick up all nine gold coins, even after the two doors have been locked?



IMPORTANT CHANGES TO THE PRIZE MATHS QUIZ!

YOU MAY NOW DISCUSS THIS PMQ PROBLEM IN THE COMMENTS BELOW AND GIVE YOUR ANSWERS AND SOLUTION METHOD.

I SHALL POST A GUIDED SOLUTION ON THE FOLLOWING FRIDAY AT THE SAME TIME AS THE NEW PMQ. AS THERE ARE OFTEN MULTIPLE WAYS OF ARRIVING AT THE SAME ANSWER, I SHALL ALSO GIVE CREDIT AND LINK TO THE BEST ANSWERS SUBMITTED IN THE COMMENTS.

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How to Enter

IF YOU ARE A STUDENT, you may still send your complete solution by email to pmq29cauchy@giftedmaths.com. This email address shall be removed after the competition closes to avoid spam. This PMQ29 competition closes on WEDNESDAY 31 JULY at 23:59 GMT. 

The Prize

The prizes for this PMQ are 3 free places in our Online Maths Club for ONE YEAR. 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. All winners can have their name posted and a link to their own online profile at their favourite social network or their own blog.

Quick Rules

Look at the expanded rules on our PMQ page.
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