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If you know the rules really well, now I give you enough time to solve this pattern: (it takes 76 moves!)
Red, Red, Red, Purple, Yellow, Yellow, Purple, Purple, Green

This is the Guinness Record of the pattern whose solution with the longest move sequence possible!

However, it is possible to make:

through this↓

Now I want you to solve another pattern:
Yellow, Red, Green, Purple, Red, Yellow, Purple, Yellow, Red

Guess how I made it this bonus puzzle: (it takes 35 moves!)

Believe it or not, this takes as few as 12 moves!

Code = 2^2 * 3 * 13 * 17 * 41 * 577 * 164513

(Simplified version of my deduced rules)

Arj Fvzcyvsvrq Ehyrf(!):

1. Lryybj sybjre jvyy tebj vs abar be zber guna bar pbaqvgvba vf erdhverq.

2. Vs gurer vf nal fcnpr (tnc) gung fheebhaqrq ol sybjref, terra sybjre jvyy tebj obgu urer naq orgjrra gur tncf, hayrff gurer vf gur bayl tnc urer.

3. Vs gur fryrpgrq fcnpr vf orgjrra bar fcnpr naq bar sybjre, checyr sybjre jvyy tebj. Gur sybjre arkg gb vg jvguref naq ba gur bgure fvqr vg tebjf gur fnzr nf gur bar gung jvguref.

4. Vs gurer vf ng yrnfg gjb fnzr sybjref arkg gb vg va n ebj, erq sybjre jvyy tebj. Gur sybjre ebj arkg gb vg jvguref.

move_index = [2, 4, 5, 3, 4, 3, 0, 1, 3, 4, 2, 6, 7, 1, 8, 7, 8]

The longest red row

The longest purple flower row: 

Center 2*red: try 14343434

Center 2*red and purple: try 237623434

NjM0MzQzNDMxIDQ3Njc2NzYyNzkgNjM3Mg==

(I tried another solution without planting three adjacent identical flowers in one move)

gjb sbhe guerr bar fvk rvtug gjb frira avar fvk guerr frira gjb

(the shortest solution proved by program)

There are actually mathematical proofs why the last puzzle has no solution.

In the last puzzle, the character must not touch the starting red square (called R) after starting, and must end on a square which is an even number far from R. Since the square under R (called B) and the square above R (called A) must be touched twice, and the character must not finally end on A or B, the squares next to either square must be touched 3 times (in->out->in->out). 

Thus, the connected parts (including the start) next to square A and B must be touched at least altogether (2+1)*2=6 times! That's impossible! The three squares next to A and B can only suffice 2+2+1=5 passes! (R is only touched once in the very beginning!)

nearly got it, but the mindset is desirable. Moves:
2431 68273 2978 or 2431 68273 2976

(Already Modified)

Ehyrf:

1. Lryybj sybjre jvyy tebj vs bgure pbaqvgvbaf ner abg erdhverq.

2. Vs gurer vf nal fcnpr (tnc) gung fheebhaqrq ol sybjref, gur fvghngvba vf pnyyrq Tnc Fprar, bgurejvfr Abezny Fprar.

3. Va Tnc Fprar, terra sybjre jvyy tebj obgu urer naq orgjrra gur tncf, hayrff gurer vf rknpgyl bar fcnpr be gjb fnzr sybjref arkg gb vg.

4. Va Abezny Fprar, checyr sybjre jvyy tebj vs gurer vf n sybjre arkg gb vg. Gur sybjre arkg gb vg jvguref naq ba gur bgure fvqr vg tebjf gur fnzr nf gur bar gung jvguref.

5. Va Abezny Fprar, erq sybjre jvyy tebj vs gurer vf ng yrnfg gjb fnzr sybjre arkg gb vg va n ebj, hayrff gurer vf bar fcnpr ba gur bgure fvqr arkg gb vg. Gur sybjre ebj arkg gb vg jvguref.

(There are very few exceptions on rule 3)

I figured out the rules, so I could probably code a program to solve it. And if you feel like counting all possible solutions, a program is needed if all rules are coded in it.

I made it while depicting the number "1" more clearly as a vertical line: 

77

Well, 77 might be the most optimal. I was wrong.

got the optimal solution

all ten digits ...

63000, all but one

Who could get the smallest solution and prove it?

Oh, a meow oysters. I cannot whelp replaying it cite.

Rot13 hint for the elephant solution of 3 - Vac Rescue:

Xbj gbj gb gur pyrnare gb nqwhfg lbhe ubevmbagny cbfvgvba, naq gura zbir fvqrjnlf gb gur qrfpraqvat pyrnare. Znxr fher lbh ner ba gur evtug cneg bs gur ebbz.

SSBoYXZlIG90aGVyIGFuc3dlcnMgZXhjZXB0IGZvciBub29kbGUobXkgZmlyc3QgYW5zd2VyKToKCkRlZXBlbiwgc3RldmVucywgc2lnaGluZywgaXRjaGluZywgZmlzaGluZywgdHVpdGlvbiwgbmVzdGVycywgd2VzdGVybiwgbWVuZGVycywgYXV0dW1uKCopLCAuLi4=

About my base64 coded thoughts about the game

you should not turn shallow with something bred

You can only get 123, 14, 24, 34 at the top, you cannot get 124, 134 or 234. But you can get 4 at the top pond and 12 at the second pond.

that's impossible

Well, there is even a method to collect all mangoes without retrying/pressing the key X!