A few weeks ago I was introduced to a simple and intriguing game. It is probably best categorised as a mathematical puzzle, yet it requires no calculations at all, just placement of numbers based on two rules. Hence, I prefer to refer to this puzzle as a pattern recognition game, an exercise for the mind using an open maze – a meditation of sorts.
The Purpose of the Game
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To take a 10 by 10 grid, representing 100 squares, and completely fill every square based on two types of movements.
The Rules are Simple
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1) Create a 10 by 10 grid (graph paper or a spread sheet will come in very handy).
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2) Fill one of the squares with the number 1, then, based on the following two movement types, use consecutive numbers to fill in the rest of the grid:
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Movement Type I) If the next number in the sequence is going to be placed vertically or horizontally, then it must be placed exactly three squares away from the previous number (there must be a two square gap between the numbers).
Movement Type II) If the next number in the sequence is going to be placed diagonally, then it must be placed exactly two squares away from the previous number (there must be a one square gap between the numbers).
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3) Numbers must be placed using the above two movement types. Numbers can only be placed in empty squares - once a square has been used it can not be used again. All placements must occur within the grid.
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4) Game is complete when the number 100 is reached - when all empty squares in the grid are filled. If you can not continue placing numbers based on the two movement types above, and empty squares remain in the grid, then the puzzle is not complete and the game lost.
Two Incomplete Examples and a Solution
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Example 1) The first example is one where I tried to get the worst possible result. I was able to end the game in 30 moves – reaching the number 30.
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Example 2) The second example is my best result so far. In this example I was able to reach 99 – one move short of finishing the game.
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A Solution: The final example is a solution completed by someone who has been playing this game for approximately three years. This is the only solution that he has ever been able to obtain for this game during this period - so we know that there is at least one solution. It would be interesting to find out how many solutions actually do exist – any hardcore mathematicians or programmers out there interested in tackling this problem?
Variations
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Other than creating a scoring system or using symbols instead of numbers to fill in the squares, two of the best variants would be: 1) to give a starting point and an end point for the game – this should increase the difficulty of the puzzle; 2) creating different types of grids, but maintaining the same placement rules – an example is presented below (click image to enlarge).
For anyone who wants to try this out, below you will find a sheet with 12 grids (click to enlarge). Ideally, a simple program based on the parameters above loaded on a device with a touchscreen would be the best way to enjoy this game.
If you are into such things, I hope the game entertains you, and if you find anymore solutions, then please send them my way. Thank you in advance.
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i just checked out some of the info on your company D-Wave Systems ... what a great project. This is what will most likely kick in the next stage in our evolution, both as a society and personally.
Once Moore's law tapers off, quantum computing should kick start things again... i hope so anyway.
For sure, if you can, please email me what you couldn't post in the comment and i'll see what i can do to put together a piece to share the info (if that's okay that is). I'm not a programmer but i know there are regular chycho readers that would appreciate it.
please include any info about your company and the project that you would like shared. I'll do my best to put the info together coherently (as best as i can understand it anyway).
you can email me at chycho@chycho.com
and thanks again for taking the time to share the knowledge :)