Fifty Flips

Note that this is NOT an "unfair" or even "EXTREMELY unfair" coin. It's not a fixed bias or even a wear pattern or other understandably-moving probability.  There is no possible way for a real coin to have that distribution.

It would be a more interesting prediction challenge if the range of posssible patterns were known.  "I've coded something that doesn't make sense; go!" is not that helpful.  Or even just say "predict the output of this program, which is absolutely not a coin".

This would be more to my taste (I can't speak for anyone else's) if we were told more about the space of possible unfairnesses. In particular, it wasn't clear to me whether

nor whether

Again, I don't claim that my taste is anyone else's; but my reaction to the extreme open-endedness is along the lines of "this could be practically anything, and some varieties of thing-it-could-be are obviously not deducible with any confidence from 50 bits of information, and this has probably been designed so that it's solvable if you correctly guess what space of possibilities the creator had in mind but I don't feel like trying to read his mind".

The spoiler-blocks above aren't very spoilery since they are just asking questions. But for the benefit of anyone who feels the same way as I do, here are what I now believe to be the answers to those questions:

You link to index C twice, rather than linking to index D. (And index D was such an interesting one too.)

Anyways, this is very fun. I made a couple (fairly easy) coins of my own, should be runnable by pasting into the console in your browser's dev tools (while you're on the fifty flips page, of course):

eval(unescape(escape`𩡬𪑰𫡯🐰𞱣𫱲𬡥𨱴𫡯🐰𞱰𬡥𩁩𨱴𩑤𬰽𦱝𞱡𨱴𭑡𫁳👛𧐻𩡵𫡣𭁩𫱮𘁦𫁩𬀨𬁲𩑤𪑣𭁥𩀩𮱩𩠨𩡬𪑰𫡯🀵𜀩𮱦𫁩𬁮𫰫🐱𞱩𩠨𣑡𭁨𛡲𨑮𩁯𫐨𚐼𩡬𪑰𫡯𛰵𜀮𜀩𮱡𨱴𭑡𫀽𘡔𨑩𫁳𘡽𩑬𬱥𮱡𨱴𭑡𫀽𘡈𩑡𩁳𘡽𬁲𩑤𪑣𭁥𩁳𛡰𭑳𪀨𬁲𩑤𪑣𭁥𩀩𞱡𨱴𭑡𫁳𛡰𭑳𪀨𨑣𭁵𨑬𚐻𪑦𚁰𬡥𩁩𨱴𩑤🐽𨑣𭁵𨑬𚑻𨱯𬡲𩑣𭁮𫰫🐱𯑤𫱣𭑭𩑮𭀮𩱥𭁅𫁥𫑥𫡴𠡹𢑤𚀢𬡥𬱵𫁴𬰢𚐮𪑮𫡥𬡈𥁍𣀽𚀢𣱮𘁦𫁩𬀠𘠫𩡬𪑰𫡯𚰢𛀠𮑯𭐠𬁲𩑤𪑣𭁥𩀠𘠫𬁲𩑤𪑣𭁥𩀫𘠻𘁴𪁥𘁣𫱩𫠠𨱡𫑥𘁤𫱷𫠠𘠫𨑣𭁵𨑬𚰢𛠢𚐻𩁯𨱵𫑥𫡴𛡧𩑴𡑬𩑭𩑮𭁂𮑉𩀨𘡳𨱯𬡥𘠩𛡩𫡮𩑲𢁔𣑌🐨𘡙𫱵𘁧𭑥𬱳𩑤𘀢𚱣𫱲𬡥𨱴𫡯𚰢𛰢𚱦𫁩𬁮𫰫𘠠𨱯𬡲𩑣𭁬𮐮𘠩𞱤𫱣𭑭𩑮𭀮𩱥𭁅𫁥𫑥𫡴𠡹𢑤𚀢𬡥𨱯𬡤𘠩𛡩𫡮𩑲𢁔𣑌𚰽𚀢🁢𬠾𘠫𩡬𪑰𫡯𚰢𛀢𚱰𬡥𩁩𨱴𩑤𚰢𛀢𚱡𨱴𭑡𫀩𞱩𩠨𩡬𪑰𫡯🐽𝐰𚑻𩁯𨱵𫑥𫡴𛡧𩑴𡑬𩑭𩑮𭁂𮑉𩀨𘡴𪁥𤡵𫁥𘠩𛡩𫡮𩑲𢁔𣑌🐧𥁨𩐠𬡵𫁥𘁷𨑳𞠠🁢🠢𥁨𩐠𬁲𫱢𨑢𪑬𪑴𮐠𫱦𘁴𨑩𫁳𘁩𬰠𩡬𪑰𘰠𛰠𝐰𛠢🀯𨠾𙱽𯑽`.replace(/u../g,'')))

eval(unescape(escape`𩡬𪑰𫡯🐰𞱣𫱲𬡥𨱴𫡯🐰𞱰𬡥𩁩𨱴𩑤𬰽𦱝𞱡𨱴𭑡𫁳👛𧐻𩡵𫡣𭁩𫱮𘁦𫁩𬀨𬁲𩑤𪑣𭁥𩀩𮱩𩠨𩡬𪑰𫡯🀵𜀩𮱦𫁩𬁮𫰫🐱𞱩𩠨𩡬𪑰𫡯🐽𝐰𚑻𨑣𭁵𨑬🐢𣡯𬁥𘡽𩑬𬱥𮱡𨱴𭑡𫀽𬁲𩑤𪑣𭁥𩁽𬁲𩑤𪑣𭁥𩁳𛡰𭑳𪀨𬁲𩑤𪑣𭁥𩀩𞱡𨱴𭑡𫁳𛡰𭑳𪀨𨑣𭁵𨑬𚐻𪑦𚁰𬡥𩁩𨱴𩑤🐽𨑣𭁵𨑬𚑻𨱯𬡲𩑣𭁮𫰫🐱𯑤𫱣𭑭𩑮𭀮𩱥𭁅𫁥𫑥𫡴𠡹𢑤𚀢𬡥𬱵𫁴𬰢𚐮𪑮𫡥𬡈𥁍𣀽𚀢𣱮𘁦𫁩𬀠𘠫𩡬𪑰𫡯𚰢𛀠𮑯𭐠𬁲𩑤𪑣𭁥𩀠𘠫𬁲𩑤𪑣𭁥𩀫𘠻𘁴𪁥𘁣𫱩𫠠𨱡𫑥𘁤𫱷𫠠𘠫𨑣𭁵𨑬𚰢𛠢𚐻𩁯𨱵𫑥𫡴𛡧𩑴𡑬𩑭𩑮𭁂𮑉𩀨𘡳𨱯𬡥𘠩𛡩𫡮𩑲𢁔𣑌🐨𘡙𫱵𘁧𭑥𬱳𩑤𘀢𚱣𫱲𬡥𨱴𫡯𚰢𛰢𚱦𫁩𬁮𫰫𘠠𨱯𬡲𩑣𭁬𮐮𘠩𞱤𫱣𭑭𩑮𭀮𩱥𭁅𫁥𫑥𫡴𠡹𢑤𚀢𬡥𨱯𬡤𘠩𛡩𫡮𩑲𢁔𣑌𚰽𚀢🁢𬠾𘠫𩡬𪑰𫡯𚰢𛀢𚱰𬡥𩁩𨱴𩑤𚰢𛀢𚱡𨱴𭑡𫀩𞱩𩠨𩡬𪑰𫡯🐽𝐰𚑻𩁯𨱵𫑥𫡴𛡧𩑴𡑬𩑭𩑮𭁂𮑉𩀨𘡴𪁥𤡵𫁥𘠩𛡩𫡮𩑲𢁔𣑌🐧𥁨𩐠𬡵𫁥𘁷𨑳𞠠🁢🠢𥁨𪑳𘁣𫱩𫠠𪑳𘁡𘁨𩑬𬁦𭑬𘁣𪁥𨑴𩑲𛀠𨑮𩀠𨑬𭱡𮑳𘁬𨑮𩁳𘁴𪁥𘁳𨑭𩐠𭱡𮐠𮑯𭐠𬁲𩑤𪑣𭁥𩀮𘠼𛱢🠧𯑽𯐊`.replace(/u../g,'')))

I assumed that if you are flipping a coin, trials would be independent events and each flip would have a fixed rule (which is what happens when you flip a single coin). Instead, the coin had a different rule for odd and even numbered flips. I think that the language of the website should be amended to reflect this.

Do you mean the coin has a fixed probability of heads for each of the 50 flips (perhaps drawn from some distribution at the beginning of the game)?

Or can the probability change for each flip based on some more complicated rule?

I liked this! The game was plenty interesting and reasonably introduced. It's a fun twist on induction games with the addition of reasoning over uncertainty rather than exactly guessing a rule, though it does have the downside the relatively small number of samples can make the payoff dominated by randomness.

To offer one small piece of constructive advice on the execution, I did wish the flip history autoscrolled to the newest entry.