Unique Trees
A puzzle game about rooted trees. (v1.1)
Designed for full-screen on web browser or itch app. You can also play on Android if your screen is large enough to tap small objects, by downloading and installing the .apk file.
Note: This game does not save. Save screenshots instead!
CONTROLS:
Left-click to grow and cut trees.
INSTRUCTIONS: (click to reveal)
- The goal is to create trees of the given sizes.
- A new tree will unlock when ALL current trees are "valid" (white petals).
- Each tree must be unique. If it contains a previous tree, that part will wither and needs to be changed.
Created for the Confounding Calendar 2024.
[When this was first released, you could actually cheese the game with a simple algorithm that went on forever, because this game used a different uniqueness requirement than Weak Tree. I believe the algorithm is now correct ("inf-embedding") -- if not, it is as least very close!]
Development log
- v1.1 - Cheese removed (inf-embedding?)25 days ago
Comments
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In the playthrough I just did with a friend, I was surprised by how much the challenge ramped up at Tree 12!
I'm not sure of the math involved, but I've found that it's possible to draw Tree 13 in the final challenge and it's a straight line in 13(if you just stop at Tree 12 there can be several possibilities)
they changed the thing! see if you can beat the new version!
I wonder how possible this is, considering the original algorithm allows you to use less nodes. I wonder how much that might inflate the numbers, or alternatively how much requiring you to use all the nodes might make ceritian configurations impossible. have you tested it? (I know you tested the original version)
Yep! I tested it right before uploading. I expect that Ultra is very difficult now, but I didn't find it too difficult when using the classic tree(3) solution as a starting point.
do you mean TREE(3)? the funny thing about this game is that you can't really search up a solution, because so few people talk about the weak tree function, which I guess is what this game is bringing awareness to.
Here's the solution for weak tree(3) that helped me find a similar solution in my game: https://math.stackexchange.com/questions/3837501/before-the-current-lower-bound-... (Oh, and I should also credit my younger brother, who told me about this problem in the first place. He was already familiar with this solution, which helped me test the game initially.)
True, but I believe that this is equivalent to weak tree(3) -- just ignore the trunks! (However, there may be some slight differences between the two once I use the proper inf-embedding algorithm [for determining uniqueness].)
[edit: To clarify, the trunk on the big tree and the subtree essentially cancel out, because every non-trunk branch has a parent branch. So if we removed trunks from all the trees, nothing would change.]
At this point it can still be cheesed, because e.g. the game incorrectly allows N(L,N(L,L)) to predate N(L,N(N(L,L))), where N is an inner node and L is a leaf. I think the correct inf-embedding allows for embeddings with gaps as long as the order of tree nodes is preserved, but now the game only checks for contiguous subtrees, which is strictly stronger.
The correct embedding checking should be easy to implement, but to check embedding of T into S, each branching node in T will create a possibility of permutation, together with all the backtracking, I guess will blow up the complexity?
really cool idea! i love how much thinking is involved with such a simple puzzle iteration
Nice little math problem, reminds me of the fun bits of math class. Also looks nice. The base game was very solvable but I wasn't expecting the ultra challenge; that really forces you to think about the algorithm.
Why does it feel weird to say that I "100%'d" a Confounding Calendar game? lol
at the beginning i didn’t know about the bonus challenge so i was about to say it was a bit easy, but the bonus challenge was tricky to solve. enjoyed it, and i didn’t really know about what the tree functions were before because the explanations on wikipedia had too much mathematical jargon. this is a good way to explain them
I also greatly this introduction to this topic. perhaps this will allow more people to become involved in and learn about this.
the trees curve too much, it can go offscreen or into adjacent frames
nevermind, you can change the direction by adding and removing branches.
--Spoilers ahead--
As far as I can tell, this can go on infinitely? I know the tree function is not infinite, so there must be a difference between the two. You can see my possibly infinite solution below:
very smart!
the tree(3) problem introduces colors of seeds (different colors of flowers) and also counts additional cases. like if the second tree in your series blue tips and red for the rest, then it would be forbidden for the next one to have blue tips and a red "trunk". imagine it removed the flower in the middle. this type of similar structure would also not be allowed. also remember that it is true in both cases that order maters, ie. it matters which way along a branch is "up".
ah thanks I remember the distinction now
the TREE function also starts with one node (so not even the trunk)
this game is a relaxed version of tree(4) edit: by the game i mean the ultra challenge
note:
TREE(n) starts with one node and you have n colors
tree(n) starts with n+1 nodes and you have 1 color
sorry I only knew the strong tree function, that is what this is a description of.
Nice!
I've completed the ultra challenge by just applying the same algorithm i used in base four. Although I probably misunderstood the trivia question
It was my mistake. Currently, the trivia question's actual answer is infinite, of course.
The infinite algorithm was unintended, resulting from the difference in uniqueness rules between this game and the Weak Tree Function. I'm hoping to patch it out later today. [edit: Finally fixed it, about a week later.]
Oh, so it'll be two puzzles today 😊
Sorry for the delay, it's fixed now. (A couple days ago, actually.)