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Here is something I really have tried to figure out,
but I am stumped by this problem.
Here it is (everything in integers):
1) There is two "sets"; A and B. Both are increasing in
the same way (we know the increasing).
2) B starts at a place on A (which we also know).
3) Sooner or later, they will share a common point (here called F).
Of course, there is infinite solutions, but it is all about
to find the FIRST one.
Let's look at a scenario with some examples.
*************************************************
*************************************************
Scenario: The points in the two "sets" increases such, that
Point= X^2, so
If X= 0, 1, 2, 3, 4,... then
Points= 0, 1, 4, 9, 16,...
- - - - - - - - - - - - - - - - - - - - - - - - -
Example 1: B starts at A= 7
...01..4...9...16
A: I I--I--*-I------I
B: ...I I--I----I
.....01..4...9
Their common point (F) is reached at A= 16
- - - - - - - -
Example 2: B starts at A= 11
...01..4...9...16...25...36
A: I I--I----I-*----I--------I----------I
B:....I I--I----I------I--------I
......0.1..4...9...16...25
Their common point (F) is reached at A= 36
- - - - - - - -
Example 3: B starts at A= 13
...01..4...9...16...25...36...49
A: I I--I----I---*--I--------I----------I------------I
B:......I I--I----I------I--------I----------I
.....0.1..4...9...16...25...36
Their common point (F) is reached at A= 49
*************************************************
*************************************************
The formula for the scenario is: Start + K^2 = L^2
Here is a table showing the FIRST solution for every
"starting point" value. It goes like this:
Starting point...Common point...Start+K^2=L^2
-----------------...-------------...-----------------
B starts at A=3:...B= 1, A= 4 -->...3 + 1 = 2
B starts at A=5:...B= 4, A= 9 -->...5 + 2 = 3
B starts at A=7:...B= 9, A= 16 -->...7 + 3 = 4...(Example 1)
B starts at A=8:...B= 1, A= 9 -->...8 + 1 = 3
B starts at A=9:...B=16, A= 25 --> ...9 + 4 = 5
B starts at A=11:...B=25, A= 36 -->...11 + 5 = 6...(Example 2)
B starts at A=12:...B= 4, A= 16 -->...12 + 2 = 4
B starts at A=13:...B=36, A= 49 -->...13 + 6 = 7...(Example 3)
B starts at A=15:...B= 1, A= 16 -->...15 + 1 = 4
B starts at A=16:...B= 9, A= 25 --> ...16 + 3 = 5
B starts at A=17:...B=64, A= 81 -->...17 + 8 = 9
B starts at A=19:...B=81, A=100 -->...19 + 9 = 10
B starts at A=20:...B=16, A= 36 -->...20 + 4 = 6
B starts at A=21:...B= 4, A= 25 -->...21 + 2 = 5
B starts at A=23:...B121, A=144 -->...23 + 11 = 12
B starts at A=24:...B= 1, A= 25 -->...24 + 1 = 5
B starts at A=25:...B144, A=169 -->...25 + 12 = 13
B starts at A=27:...B= 9, A= 36 -->...27 + 3 = 6
B starts at A=28:...B=36, A= 64 -->...28 + 6 = 8
B starts at A=29:...B=196, A=225 -->...29 + 14 = 15
B starts at A=31:...B=225, A=256 -->...31 + 15 = 16
B starts at A=32:...B= 4, A=36 -->...32 + 2 = 6
B starts at A=33:...B=16, A=49 -->...33 + 4 = 7
B starts at A=35:...B=289, A=324 -->...35 + 17 = 18
B starts at A=36:...B=64, A=100 -->...36 + 8 = 10
B starts at A=37:...B=324, A=361 -->...37 + 18 = 19
B starts at A=39:...B=25, A=64 -->...39 + 5 = 8
B starts at A=40:...B= 9, A=49 -->...40 + 3 = 7
B starts at A=41:...B=400, A=441 -->...41 + 20 = 21
B starts at A=43:...B=441, A=484 -->...43 + 21 = 22
B starts at A=44:...B=100, A=144 -->...44 + 10 = 12
B starts at A=45:...B= 4, A=49 -->...45 + 2 = 7
B starts at A=47:...B=529, A=576 -->...47 + 23 = 24
B starts at A=48:...B=16, A=64 -->...48 + 4 = 8
B starts at A=49:...B=576, A=625 -->...49 + 24 = 25
As we see, the answers ("Common point") are NOT
linear, because they are based on the fact that I want
the first common point...
It would be interesting to have a formula for solving this!
Thanks in advance!
but I am stumped by this problem.
Here it is (everything in integers):
1) There is two "sets"; A and B. Both are increasing in
the same way (we know the increasing).
2) B starts at a place on A (which we also know).
3) Sooner or later, they will share a common point (here called F).
Of course, there is infinite solutions, but it is all about
to find the FIRST one.
Let's look at a scenario with some examples.
*************************************************
*************************************************
Scenario: The points in the two "sets" increases such, that
Point= X^2, so
If X= 0, 1, 2, 3, 4,... then
Points= 0, 1, 4, 9, 16,...
- - - - - - - - - - - - - - - - - - - - - - - - -
Example 1: B starts at A= 7
...01..4...9...16
A: I I--I--*-I------I
B: ...I I--I----I
.....01..4...9
Their common point (F) is reached at A= 16
- - - - - - - -
Example 2: B starts at A= 11
...01..4...9...16...25...36
A: I I--I----I-*----I--------I----------I
B:....I I--I----I------I--------I
......0.1..4...9...16...25
Their common point (F) is reached at A= 36
- - - - - - - -
Example 3: B starts at A= 13
...01..4...9...16...25...36...49
A: I I--I----I---*--I--------I----------I------------I
B:......I I--I----I------I--------I----------I
.....0.1..4...9...16...25...36
Their common point (F) is reached at A= 49
*************************************************
*************************************************
The formula for the scenario is: Start + K^2 = L^2
Here is a table showing the FIRST solution for every
"starting point" value. It goes like this:
Starting point...Common point...Start+K^2=L^2
-----------------...-------------...-----------------
B starts at A=3:...B= 1, A= 4 -->...3 + 1 = 2
B starts at A=5:...B= 4, A= 9 -->...5 + 2 = 3
B starts at A=7:...B= 9, A= 16 -->...7 + 3 = 4...(Example 1)
B starts at A=8:...B= 1, A= 9 -->...8 + 1 = 3
B starts at A=9:...B=16, A= 25 --> ...9 + 4 = 5
B starts at A=11:...B=25, A= 36 -->...11 + 5 = 6...(Example 2)
B starts at A=12:...B= 4, A= 16 -->...12 + 2 = 4
B starts at A=13:...B=36, A= 49 -->...13 + 6 = 7...(Example 3)
B starts at A=15:...B= 1, A= 16 -->...15 + 1 = 4
B starts at A=16:...B= 9, A= 25 --> ...16 + 3 = 5
B starts at A=17:...B=64, A= 81 -->...17 + 8 = 9
B starts at A=19:...B=81, A=100 -->...19 + 9 = 10
B starts at A=20:...B=16, A= 36 -->...20 + 4 = 6
B starts at A=21:...B= 4, A= 25 -->...21 + 2 = 5
B starts at A=23:...B121, A=144 -->...23 + 11 = 12
B starts at A=24:...B= 1, A= 25 -->...24 + 1 = 5
B starts at A=25:...B144, A=169 -->...25 + 12 = 13
B starts at A=27:...B= 9, A= 36 -->...27 + 3 = 6
B starts at A=28:...B=36, A= 64 -->...28 + 6 = 8
B starts at A=29:...B=196, A=225 -->...29 + 14 = 15
B starts at A=31:...B=225, A=256 -->...31 + 15 = 16
B starts at A=32:...B= 4, A=36 -->...32 + 2 = 6
B starts at A=33:...B=16, A=49 -->...33 + 4 = 7
B starts at A=35:...B=289, A=324 -->...35 + 17 = 18
B starts at A=36:...B=64, A=100 -->...36 + 8 = 10
B starts at A=37:...B=324, A=361 -->...37 + 18 = 19
B starts at A=39:...B=25, A=64 -->...39 + 5 = 8
B starts at A=40:...B= 9, A=49 -->...40 + 3 = 7
B starts at A=41:...B=400, A=441 -->...41 + 20 = 21
B starts at A=43:...B=441, A=484 -->...43 + 21 = 22
B starts at A=44:...B=100, A=144 -->...44 + 10 = 12
B starts at A=45:...B= 4, A=49 -->...45 + 2 = 7
B starts at A=47:...B=529, A=576 -->...47 + 23 = 24
B starts at A=48:...B=16, A=64 -->...48 + 4 = 8
B starts at A=49:...B=576, A=625 -->...49 + 24 = 25
As we see, the answers ("Common point") are NOT
linear, because they are based on the fact that I want
the first common point...
It would be interesting to have a formula for solving this!
Thanks in advance!