February 2020

[Dieter]

You are right, but it is notable that the one who has the bonus always is a fast solver, often with *. Either luck or many threads?
Or there is a clue?

[LaurV]

We are, for sure, missing something here. We played with this for a couple of hours this wonderful Saturday morning, in front of a mug with coffee, but we can't get close to any number you guys talk about... We think that our understanding of the puzzle is probably wrong.

Giving up (for now), and moving to do some real work. Good luck to all who are stubborn enough to hunt for solutions and stars!

[Dieter]

Originally Posted by yae9911

Thanks!

Well then I don't have to worry about it anymore. I got both with the random simulation, with which I can recalculate the original game quite accurately, and with an exact calculation the approx. 33.333 ...
The exact result should be

77793808048991155069512637767746406705805011749411165293240199952210986407 /
 2333814241469732031952625840042216151324387397379954245052697639351484416
 =  33.33333333333337075608827723...

Just for fun and comparison: Do you have more digits after the point? I have computed the case without ladders and snakes using octuple-precision floats (256 bit).
You have:

33.33333333333337075608827723...

My digits:

33,33333333333337075608827723077517516365419018158718837413...

(Auf die schriftliche Division mit Papier und Bleistift kann ich verzichten!)

[Flow]

Originally Posted by SmartMersenne

There are ~100²⁰ possible combinations.

It is like the puzzle-master is saying "I have a combination in mind, can you find it?"

There doesn't seem to be any clue as to how to find it. And we all have been trying for the last 2 weeks to find it by random search.

Good luck!

Except that the exponential number of solutions is not a proof of complexity.

@Deiter 256 decimals
33.3333333333333707560882772307751751636541901815871883741305500801482\
1204666202969926958266548184843891480339338552289078348822180513075920\
6678407482876919981059403231292325465970228085296008800508876654103495\
45028425962346565458075668868724379723599923411

[SmartMersenne]

Originally Posted by Flow

Except that the exponential number of solutions is not a proof of complexity.

@Deiter 256 decimals
33.3333333333333707560882772307751751636541901815871883741305500801482\
1204666202969926958266548184843891480339338552289078348822180513075920\
6678407482876919981059403231292325465970228085296008800508876654103495\
45028425962346565458075668868724379723599923411

It is not complex, actually, it is pretty simple:

1) generate a random configuration
2) Test it
3) If no solution found go to 1

Unfortunately, the solution space is extremely sparse, which makes it challenging to find the right configuration.

[Dieter]

Originally Posted by Flow

Except that the exponential number of solutions is not a proof of complexity.

@Deiter 256 decimals
33.3333333333333707560882772307751751636541901815871883741305500801482\
1204666202969926958266548184843891480339338552289078348822180513075920\
6678407482876919981059403231292325465970228085296008800508876654103495\
45028425962346565458075668868724379723599923411

Thank you! My first 67 digits after the point are the same - applying the formulas of Althoen etc. using 256 bit floats I cannot demand more.
Where do your digits come from? Have you calculated the fraction of yae9911 or have you used the formulas of Althoen etc. for the empty board?

Apart from that I get many solutions with error<5*10**(-7) and some solutions with ...*10**(-8) or ...*10**(-9).
My best combination yields 66,978705007997 (error = 4,42*10*(-10)).
Unfortunately this combination doesn‘t help at all to find better solutions.

[yae9911]

Originally Posted by SmartMersenne

It is not complex, actually, it is pretty simple:

1) generate a random configuration
2) Test it
3) If no solution found go to 1

Unfortunately, the solution space is extremely sparse, which makes it challenging to find the right configuration.

There are much better methods than random search. You can e.g. use the available 10 shortcut steps hierarchically. Part of it to roughly set the desired game length (Hint: Use feedback loops with overlapping snakes) and the remaining very short steps (e.g. of length +-1) to fine-tune the game length. It is easy to try that changing the starting positions of the short steps with a frozen length is enough to change the game's duration slightly.

[Dieter]

Originally Posted by yae9911

There are much better methods than random search. You can e.g. use the available 10 shortcut steps hierarchically. Part of it to roughly set the desired game length (Hint: Use feedback loops with overlapping snakes) and the remaining very short steps (e.g. of length +-1) to fine-tune the game length. It is easy to try that changing the starting positions of the short steps with a frozen length is enough to change the game's duration slightly.

My best configuration contains the pair [26,25]. If I change the position with frozen length I get:

[25,24] 42,265...
[26,25] 66,978705008 (error = 4,42*10**(-10))
[27,26] 66,98667..... (error = 8*10**(-3))

That are jumps, no fine-tuning is possible.

What is your best value gotten by fine-tuning?

[Xyzzy]

http://www.research.ibm.com/haifa/po...ruary2020.html

[LaurV]

Nice, hehe... Congrats to the winners!
We like the idea of halving the search space...