Alright, here's the answer to the last question: yup,


Solution:
Note: it would be an advantage if the fundamental nature of this problem is understood before we attempt to solve the higher application. Therefore, we would attempt to find the distribution of pigeons for the 1,2,3,4,5 configuration before proceeding on to main bulk of the question.

• There are 15 pigeons to be divided into five coconut trees. There are a few possibilities of distribution of pigeons:

• An increasing distribution i.e. coconut tree 1=1 pigeon, coconut tree2=2 pigeons and so on…
• Since there are only 15 pigeons in total, the only possible increasing distribution is 1, 2, 3, 4, 5. Where 1+2+3+4+5=15

• A decreasing distribution i.e. coconut tree 1=5 pigeons, coconut tree 2= 4 pigeons and so on…
• Due to the same above mentioned reason, the only possible decreasing distribution is 5, 4, 3, 2, 1.

• A random distribution i.e. coconut tree 1=2pigeons, coconut tree 2=5 pigeons etc…
• There are many possibilities for a random distribution, they may include:
• 2, 5, 1, 3, 4
• 3, 3, 4, 2, 3
• …

• or

• An equal distribution ie coconut tree 1=3 pigeons, coconut tree 2=3 pigeons and so on, where the distribution is given 3, 3, 3, 3, 3.

Do note that the dance cycle is such that after the first round, the results would generally be as follows:

• Coconut tree 1 will gain 4 pigeons
• Coconut tree 2 will lose 1 pigeon
• Coconut tree 3 will lose 1 pigeon
• Coconut tree 1 will lose 1 pigeon
• Coconut tree 5 will lose 1 pigeon

Do note that the full cycle of pigeons and trees is tree 1 to 2, then 2 to 3, then 3 to 4, then 4 to 5, and 5 to 1. Since the cycle ends at tree 1, instead of tree 5 (which intuition may lead us to think as such), and since the dance is such that no tree is repeated twice consecutively, therefore the next step moves on to tree 2.

A good visualization may be acquired if we were to place the trees in a cyclic diagram rather than in a linear diagram.

And after the second cycle,
• Coconut tree 1 will lose one pigeon in the second dance to gain 3 pigeons in total
• Coconut tree 2 will gain 4 pigeons in the second dance to gain 3 pigeons in total
• Coconut tree 3 will lose one pigeon in the second dance to lose 2 pigeons in total
• Coconut tree 4 will lose one pigeon in the second dance to lose 2 pigeons in total
• Coconut tree 5 will lose one pigeon in the second dance to lose 3 pigeons in total

Should we follow the next cycle, we will have the following outlines:

1) lose 1 in 3rd dance to gain 2 in total,
2) lose 1 in 3rd dance to gain 2 in total,
3) gain 4 in 3rd dance to gain 2 in total,
4) lose 1 in 3rd dance to lose 3 in total,
5) lose 1 in 3rd dance to lose 4 in total,

Do note that each coconut tree has it’s turn in gaining 4 pigeons (from 1st to 2nd and to 3rd), if we were to let the dance go on for another 2 iterations, we will have the outcome:

All coconut trees will gain 4 pigeons in 1 turn and lose 4 pigeons through 4 turns (1 each)

Now, the solution is to sub in the above 4 scenarios to see how the numbers turn out

Note that it is impossible to put in 3,3,3,3,3. This is because in the process, the 5th coconut tree will run out of pigeons before any it gets its turn of gaining 4 pigeons (the -1+5 turn).

This is the same with the decreasing distribution 5,4,3,2,1.

In general, each coconut tree must have enough pigeons to sustain the -1 pigeon turn before it gets its own +4 turn.

Thus the only possible configuration is 1,2,3,4,5 where there are 1 pigeon in the 1st tree, 2 pigeons in the 2nd tree, etc

With the outcome after each step as:

1, 2, 3, 4, 5
5, 1, 2, 3, 4
4, 5, 1, 2, 3
3, 4, 5, 1, 2
2, 3, 4, 5, 1
1, 2, 3, 4, 5


The essence to deduce is that the difference between the number of pigeons per tree is= the configuration of pigeon movement between the trees



Now, regarding part 2

• Note that in the 3,7,8,10,16 configuration, there are a total of 44 birds in transition per turn.

• Technically, owing to part 1, having coconut tree 1=3 pigeons, coconut tree 2=7 pigeons, coconut tree 3= 8 pigeons etc will let us come back to the configuration of 3,7,8,10,16.

• Do note that 3+7+8+10+16 is exactly 44

• And henceforth the answer being a distribution of

• 3 pigeons in the 1st tree
• 7 pigeons in the 2nd tree
• 8 pigeons in the 3rd tree
• 10 pigeons in the 4th tree
And
• 16 pigeons in the 5th tree