# Math riddle

#### Cooldefekt

##### New member
Hi people
Sorry if this riddle post is in wrong category, first time here and little time since I got interested in math.
But lets write a math riddle that got me hook since I first heard it and I know its all math behind its solution:

Interesting mathematics:
In a lake there is just one fish in the begining. That fish spawn(produce) for the first time one fish egg on the seventh day and after that on every 6th day. Every new fish spawn the same way, for the first time on seventh day and after on every sixth day. How to find how namy fish there will be in 80 days( taking account the spawning of the first fish and all the others)?

#### Subhotosh Khan

##### Super Moderator
Staff member
Hi people
Sorry if this riddle post is in wrong category, first time here and little time since I got interested in math.
But lets write a math riddle that got me hook since I first heard it and I know its all math behind its solution:

Interesting mathematics:
In a lake there is just one fish in the begining. That fish spawn(produce) for the first time one fish egg on the seventh day and after that on every 6th day. Every new fish spawn the same way, for the first time on seventh day and after on every sixth day. How to find how namy fish there will be in 80 days( taking account the spawning of the first fish and all the others)?
Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

#### Cooldefekt

##### New member
As am not mathematician, I tried by logic but dont know if am on right track and if I am is there a formula for it. Here is what I tried

#### Steven G

##### Elite Member
For the 1st fish why did you stop at day 49? Why not go to 79

1: 1 8 14 20 26 32 38 44 50 56 62 68 74 80----total of 14 fish

Now for fish born on day 8
8: 8 15 21 27 33 39 45 51 57 63 69 75 ---total of 11 new fish

How about fish born on day 14?

Day 20?

What are all the day numbers that fish can be born on?

#### Cooldefekt

##### New member
HI.
I tried and I wasn't sure if I am on the right track and I stopped on the 50th day. The riddle says all born fish till 80th day.

#### Steven G

##### Elite Member
So you have given up?

#### Steven G

##### Elite Member
Why not try some more? Show us your work and you will be given hints to proceed further,

#### JeffM

##### Elite Member
Think about fish spawned on days that are multiples of 7?

How many are there?

#### Cooldefekt

##### New member
If Im not wrong there will be 11 fish that give birth on their 7th day in 80 days?

Maybe I should try in Excel

#### Cubist

##### Senior Member
Maybe I should try in Excel
I can't think of a nice mathematical way to accomplish this in a single formula. Excel might be one way. However, here's some Python code that does the trick. It isn't optimised, but it ought to be fairly clear how it works.

Python:
maxDay=80

def newFish(fishSpawnedOnDay, day):
fishSpawnedOnDay[day] += 1
day += 7
while day < maxDay:
newFish(fishSpawnedOnDay, day);
day += 6

############

fishSpawnedOnDay = [0] * maxDay # Create an array of numbers, all initialised to 0

newFish(fishSpawnedOnDay, 0) # recursively create all the fish

# Show the results
print(" Day   #spawned");
for i in range(maxDay):
print("  ", i+1, " ", fishSpawnedOnDay[i])
print("Total:", sum(fishSpawnedOnDay))

Code:
 Day   #spawned
1   1
2   0
3   0
4   0
5   0
6   0
7   0
8   1
9   0
10   0
11   0
12   0
13   0
14   1
15   1
16   0
17   0
18   0
19   0
20   1
21   2
22   1
23   0
24   0
25   0
26   1
27   3
28   3
29   1
30   0
31   0
32   1
33   4
34   6
35   4
36   1
37   0
38   1
39   5
40   10
41   10
42   5
43   1
44   1
45   6
46   15
47   20
48   15
49   6
50   2
51   7
52   21
53   35
54   35
55   21
56   8
57   9
58   28
59   56
60   70
61   56
62   29
63   17
64   37
65   84
66   126
67   126
68   85
69   46
70   54
71   121
72   210
73   252
74   211
75   131
76   100
77   175
78   331
79   462
80   463
Total: 3535

#### JasonTurner

##### New member
The problem seems to be sequence-related. The statement refers to the fish and its reproductive cycle, fish laying an egg every sixth day, and the total number of fish in the lake after 80 days considering each fish spawn every sixth day.
The solution goes in sequence from the first fish and counts towards its reproductive cyclicity every week, where the fish lays an egg on the sixth day. A table would be an easy way to solve this situation with that case in mind. Listing the number of fish each week and calculating the total fish until the 79 days starting from day 0, the number falls to the 3535 fish in lake on day 80.
On day 0, only one fish was available, and on day 7, it laid eggs and grew to adult fish and then laid eggs on day 6th of their cycle and continued, which was followed till day 80 and the population calculated to 3535.

Last edited by a moderator:

#### lookagain

##### Elite Member
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Staff member