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Count lizards on an island

The question is, how to count the lizards on an island?

For example, 15 lizards are on the island. Each step should be handled on a day. The symbol for unmarked lizards is o and for marked lizards is O.

day Work Lizards (e.g.) markup ratio
0 the day before … ooo ooo ooo ooo ooo 0.0
1 mark each found lizard OOO OOO OOO ooo ooo 0.66666
2 count marked lizard OOOoo 0.66666

We found only 9 lizards on day 1 to mark. We don't known the number of the other unfounded lizards and we don't know the markup ratio.

During the night marked and unmarked lizards are mixed.

On day 2, we found only 5 lizards, 3 marked and 2 unmarked. Now we can calculate the markup ratio as 2/3=0.6666666666. On day 1 we marked 9 lizards what is 0.66666 of all lizards.

We can calculate to all lizards a or unmarked lizards u:

 u     2          2 * 9
--- = --- => u = ------- = 6
9     3            3       

… or …

 9     3          9 * 5
--- = --- => a = -------- = 15
a     5            3         

This method is based on the markup ratio and the mixing at night. The markup ratio is a constant fixed on day 1.

Some lizard are hidden, and we can't found them to markup. In the night some hidden lizards become visible and previously visible lizards hide. On the next day, the number of found lizards follows the markup ratio.

If there are only two species of lizards to count, the procedure also works.

day Work Lizards (e.g.) x o
0 the day before … xxx xxx ooo ooo ooo xxx xxx ooo ooo ooo
1 mark each found lizard XOO XOO XOO xoo xoo XXX xxx OOO OOO ooo
2 count marked lizard XOOxo Xx OOo
 3     1           3 * 2
--- = --- => aX = -------- = 6
aX    2             1        
 6     2           6 * 3
--- = --- => aO = -------- = 9
aO    3             2