Index within a triangular grid
I have the complete set of integer pairs ($a,b$) with the following constraints:
$a geq 0$, $b geq 0$
$a<n$, $b<n$
- $a<b$
The count of pairs will equal $(n)(n-1)/2$
If n=5, there would be ten pairs.
I am declaring the index value to be p.
| p | a,b |
|-------|-------|
| 0 | 0,1 |
| 1 | 0,2 |
| 2 | 0,3 |
| 3 | 0,4 |
| 4 | 1,2 |
| 5 | 1,3 |
| 6 | 1,4 |
| 7 | 2,3 |
| 8 | 2,4 |
| 9 | 3,4 |
This can be intuitively laid out as a triangle, with a numbering each row, b numbering each column, and p being the value in each cell.
| 1 | 2 | 3 | 4 |
+---+---+---+---+
0 | 0 | 1 | 2 | 3 |
1 | | 4 | 5 | 6 |
2 | | | 7 | 8 |
3 | | | | 9 |
There are two questions that I'm stumped on:
- Given n and an index value p, is there a formula to determine the values of a and b?
- Given n, a and b, can I determine the index p?
algebra-precalculus
asked Nov 27 at 5:32
Andrew Shepherd
1416
|
show 1 more comment
I have the complete set of integer pairs ($a,b$) with the following constraints:
$a geq 0$, $b geq 0$
$a<n$, $b<n$
- $a<b$
The count of pairs will equal $(n)(n-1)/2$
If n=5, there would be ten pairs.
I am declaring the index value to be p.
| p | a,b |
|-------|-------|
| 0 | 0,1 |
| 1 | 0,2 |
| 2 | 0,3 |
| 3 | 0,4 |
| 4 | 1,2 |
| 5 | 1,3 |
| 6 | 1,4 |
| 7 | 2,3 |
| 8 | 2,4 |
| 9 | 3,4 |
This can be intuitively laid out as a triangle, with a numbering each row, b numbering each column, and p being the value in each cell.
| 1 | 2 | 3 | 4 |
+---+---+---+---+
0 | 0 | 1 | 2 | 3 |
1 | | 4 | 5 | 6 |
2 | | | 7 | 8 |
3 | | | | 9 |
There are two questions that I'm stumped on:
- Given n and an index value p, is there a formula to determine the values of a and b?
- Given n, a and b, can I determine the index p?
algebra-precalculus
asked Nov 27 at 5:32
Andrew Shepherd
1416
I suspect that both my question title and the tag I used are inappropriate. If somebody would like to edit this, please go ahead.
– Andrew Shepherd
Nov 27 at 5:36
What is p? - you did not define it.
– Moti
Nov 27 at 6:38
It is not clear given a, b - for every n all (a, b) are determined by your discussion.
– Moti
Nov 27 at 6:41
@Moti - I'm calling the indexp. I'll add that to my table.
– Andrew Shepherd
Nov 27 at 7:02
First question: I think it is possible, because $p$ takes different value for different pairs $(a,b).$
– user376343
Nov 27 at 10:43
|
show 1 more comment
I have the complete set of integer pairs ($a,b$) with the following constraints:
$a geq 0$, $b geq 0$
$a<n$, $b<n$
- $a<b$
The count of pairs will equal $(n)(n-1)/2$
If n=5, there would be ten pairs.
I am declaring the index value to be p.
| p | a,b |
|-------|-------|
| 0 | 0,1 |
| 1 | 0,2 |
| 2 | 0,3 |
| 3 | 0,4 |
| 4 | 1,2 |
| 5 | 1,3 |
| 6 | 1,4 |
| 7 | 2,3 |
| 8 | 2,4 |
| 9 | 3,4 |
This can be intuitively laid out as a triangle, with a numbering each row, b numbering each column, and p being the value in each cell.
| 1 | 2 | 3 | 4 |
+---+---+---+---+
0 | 0 | 1 | 2 | 3 |
1 | | 4 | 5 | 6 |
2 | | | 7 | 8 |
3 | | | | 9 |
There are two questions that I'm stumped on:
- Given n and an index value p, is there a formula to determine the values of a and b?
- Given n, a and b, can I determine the index p?
algebra-precalculus
asked Nov 27 at 5:32
Andrew Shepherd
1416
I have the complete set of integer pairs ($a,b$) with the following constraints:
$a geq 0$, $b geq 0$
$a<n$, $b<n$
- $a<b$
The count of pairs will equal $(n)(n-1)/2$
If n=5, there would be ten pairs.
I am declaring the index value to be p.
| p | a,b |
|-------|-------|
| 0 | 0,1 |
| 1 | 0,2 |
| 2 | 0,3 |
| 3 | 0,4 |
| 4 | 1,2 |
| 5 | 1,3 |
| 6 | 1,4 |
| 7 | 2,3 |
| 8 | 2,4 |
| 9 | 3,4 |
This can be intuitively laid out as a triangle, with a numbering each row, b numbering each column, and p being the value in each cell.
| 1 | 2 | 3 | 4 |
+---+---+---+---+
0 | 0 | 1 | 2 | 3 |
1 | | 4 | 5 | 6 |
2 | | | 7 | 8 |
3 | | | | 9 |
There are two questions that I'm stumped on:
- Given n and an index value p, is there a formula to determine the values of a and b?
- Given n, a and b, can I determine the index p?
algebra-precalculus
algebra-precalculus
asked Nov 27 at 5:32
Andrew Shepherd
1416
asked Nov 27 at 5:32
Andrew Shepherd
1416
edited Nov 27 at 7:04
asked Nov 27 at 5:32
Andrew Shepherd
1416
asked Nov 27 at 5:32
Andrew Shepherd
1416
asked Nov 27 at 5:32
Andrew Shepherd
1416
1416
I suspect that both my question title and the tag I used are inappropriate. If somebody would like to edit this, please go ahead.
– Andrew Shepherd
Nov 27 at 5:36
What is p? - you did not define it.
– Moti
Nov 27 at 6:38
It is not clear given a, b - for every n all (a, b) are determined by your discussion.
– Moti
Nov 27 at 6:41
@Moti - I'm calling the indexp. I'll add that to my table.
– Andrew Shepherd
Nov 27 at 7:02
First question: I think it is possible, because $p$ takes different value for different pairs $(a,b).$
– user376343
Nov 27 at 10:43
|
show 1 more comment
I suspect that both my question title and the tag I used are inappropriate. If somebody would like to edit this, please go ahead.
– Andrew Shepherd
Nov 27 at 5:36
What is p? - you did not define it.
– Moti
Nov 27 at 6:38
It is not clear given a, b - for every n all (a, b) are determined by your discussion.
– Moti
Nov 27 at 6:41
@Moti - I'm calling the indexp. I'll add that to my table.
– Andrew Shepherd
Nov 27 at 7:02
First question: I think it is possible, because $p$ takes different value for different pairs $(a,b).$
– user376343
Nov 27 at 10:43
I suspect that both my question title and the tag I used are inappropriate. If somebody would like to edit this, please go ahead.
– Andrew Shepherd
Nov 27 at 5:36
I suspect that both my question title and the tag I used are inappropriate. If somebody would like to edit this, please go ahead.
– Andrew Shepherd
Nov 27 at 5:36
What is p? - you did not define it.
– Moti
Nov 27 at 6:38
What is p? - you did not define it.
– Moti
Nov 27 at 6:38
It is not clear given a, b - for every n all (a, b) are determined by your discussion.
– Moti
Nov 27 at 6:41
It is not clear given a, b - for every n all (a, b) are determined by your discussion.
– Moti
Nov 27 at 6:41
@Moti - I'm calling the index
p. I'll add that to my table.– Andrew Shepherd
Nov 27 at 7:02
@Moti - I'm calling the index
p. I'll add that to my table.– Andrew Shepherd
Nov 27 at 7:02
First question: I think it is possible, because $p$ takes different value for different pairs $(a,b).$
– user376343
Nov 27 at 10:43
First question: I think it is possible, because $p$ takes different value for different pairs $(a,b).$
– user376343
Nov 27 at 10:43
|
show 1 more comment
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I suspect that both my question title and the tag I used are inappropriate. If somebody would like to edit this, please go ahead.
– Andrew Shepherd
Nov 27 at 5:36
What is p? - you did not define it.
– Moti
Nov 27 at 6:38
It is not clear given a, b - for every n all (a, b) are determined by your discussion.
– Moti
Nov 27 at 6:41
@Moti - I'm calling the index
p. I'll add that to my table.– Andrew Shepherd
Nov 27 at 7:02
First question: I think it is possible, because $p$ takes different value for different pairs $(a,b).$
– user376343
Nov 27 at 10:43