This might be obvious, but if you are doing an exponential puzzle, a non-integer number that suggests itself is e, the exponential function. But I don't totally understand your new puzzle. How can each piece multiply the number of moves by 1.618..? How do you add .618.. moves? Is that an average over different pieces adding different numbers of moves?
That is a fair question, and your answer is correct. Actually, the factor by which the number of moves increases per additional piece CONVERGES to phi for large numbers of pieces. For smaller number of pieces, it may be a bit below or above.
It would be interesting to see a puzzle where the base of the exponential for the number of moves is closer to 1.25 or something near that, so that you can include a higher number of pieces and still have a reasonable solving time. Zigguphi is already a step in this direction.
@ What I mean is, in a phinary counter, adding 1 involves doing an un-carry. IE, we count like 0, 1, then to get 2 we note that 1 = 0.11 (un-carrying), then two = 1.11 = 10.01. Whereas in finary, we count 0, 1, then two is 11 which equals 100. Finary and phinary both use the same carry rule and both involve the golden ratio, but they are different numbering systems. Phinary assigns exactly the powers of phi to 1, 10, 100, 1000 etc., whereas finary assigns the Fibonacci numbers to 1, 10, 100, 1000 etc. Finary is pronounced like the beginning of “Fibonacci”.
@@OskarPuzzle I can’t find the word “phinary” on the Wikipedia page - could you say which section this is? It isn’t clear to me what the definition of a “move” is, but the common usage of it is that it is a discrete event. So you can only ever do an integer number of moves. I would imagine a number written as “pqr” in base b to be r + b*q + b^2*p, which wouldn’t give an integer most of the time when b = phi.
@@henryseg The Wikipedia page is about the word "exponential", which is the topic of the semantic discussion. Binary means 2^n. Ternary means 3^n. Phinary means phi^n. The word "phinary" does not have a Wikipedia page yet, as I just introduced it today. The puzzle converges to phinary for n going to infinity.
if i treat the state of the puzzles as an array of numbers, what are the rules for each? as i understand in the original the numbers go from 0 to 3, you can change the lowest number freely, and you may move between 0 and 1 or 2 and 3 only if the previous number is 3, and can only move between 1 and 2 if the previous number is 3. what's the corresponding rule for zigguphi?
I think the difference is this: for zigguphi, you can move between 2 and 3 when the previous number is 2 (instead of 3). You can still only move between 0 and 1 when the previous number is 3. This means that the state to disassemble the final piece is [2, 2 ... 2, 3]. After that, I'm pretty sure it's trivial to disassemble the remaining pieces like with the zigguflat, though it may require you to move each from 2 to 3 before sliding it out of the puzzle.
This might be obvious, but if you are doing an exponential puzzle, a non-integer number that suggests itself is e, the exponential function.
But I don't totally understand your new puzzle. How can each piece multiply the number of moves by 1.618..? How do you add .618.. moves? Is that an average over different pieces adding different numbers of moves?
That is a fair question, and your answer is correct. Actually, the factor by which the number of moves increases per additional piece CONVERGES to phi for large numbers of pieces. For smaller number of pieces, it may be a bit below or above.
@@OskarPuzzle OK, next question: Why?
It has an entire additional piece, as well, that's the clearly visible change ;P
What other Ziggu* puzzle can we design with a non-integer exponential base factor?
Zigguplop, ziggutrap and ziggudut comes to mind. :-)
It would be interesting to see a puzzle where the base of the exponential for the number of moves is closer to 1.25 or something near that, so that you can include a higher number of pieces and still have a reasonable solving time. Zigguphi is already a step in this direction.
ziggupi ?
I don't know what number that is, but make a 2.5D version (i.e. layered) and call it ziggu rat.
We could pedantically call this “finary” rather than “phinary” - it implements Fibonacci counting not true phinary numbers.
Wikipedia says phi: en.wikipedia.org/wiki/Golden_ratio
@ What I mean is, in a phinary counter, adding 1 involves doing an un-carry. IE, we count like 0, 1, then to get 2 we note that 1 = 0.11 (un-carrying), then two = 1.11 = 10.01. Whereas in finary, we count 0, 1, then two is 11 which equals 100.
Finary and phinary both use the same carry rule and both involve the golden ratio, but they are different numbering systems. Phinary assigns exactly the powers of phi to 1, 10, 100, 1000 etc., whereas finary assigns the Fibonacci numbers to 1, 10, 100, 1000 etc.
Finary is pronounced like the beginning of “Fibonacci”.
@@OskarPuzzle I can’t find the word “phinary” on the Wikipedia page - could you say which section this is?
It isn’t clear to me what the definition of a “move” is, but the common usage of it is that it is a discrete event. So you can only ever do an integer number of moves. I would imagine a number written as “pqr” in base b to be r + b*q + b^2*p, which wouldn’t give an integer most of the time when b = phi.
@@henryseg The Wikipedia page is about the word "exponential", which is the topic of the semantic discussion. Binary means 2^n. Ternary means 3^n. Phinary means phi^n. The word "phinary" does not have a Wikipedia page yet, as I just introduced it today. The puzzle converges to phinary for n going to infinity.
Do all positive algebraic numbers have a ZigguX puzzle?
if i treat the state of the puzzles as an array of numbers, what are the rules for each? as i understand in the original the numbers go from 0 to 3, you can change the lowest number freely, and you may move between 0 and 1 or 2 and 3 only if the previous number is 3, and can only move between 1 and 2 if the previous number is 3. what's the corresponding rule for zigguphi?
I think the difference is this: for zigguphi, you can move between 2 and 3 when the previous number is 2 (instead of 3). You can still only move between 0 and 1 when the previous number is 3. This means that the state to disassemble the final piece is [2, 2 ... 2, 3]. After that, I'm pretty sure it's trivial to disassemble the remaining pieces like with the zigguflat, though it may require you to move each from 2 to 3 before sliding it out of the puzzle.
@OskarPuzzle Can you confirm or correct my guess of the difference between the puzzles? Thanks and happy holidays!
It'd be great to see a pi or tau base version.
Supergolden ratio and plastic ratio have simple carry rules! Maybe not simple to actually implement though.
thank you for confusing me once again!