Ranking & Order

Almost every ranking question rests on one fact: a person counted from both ends is counted twice. So

total = position from one end + position from the other end − 1

and nearly every other rule is that formula rearranged.

1. Position from the other end = total − position + 1. In a row of 40, the 15th from the left is the 26th from the right.
2. People between two persons. Bring both to the same end first, then subtract and take away 1 more: between the 12th and the 31st there are 31 − 12 − 1 = 18.
3. Interchange. When A and B swap places, A's new position and B's old position describe the same seat — add them and subtract 1 for the total.
4. Crossed positions. If two positions counted from opposite ends add up to more than the total, the two people have crossed over; convert one of them and count the gap the other way.
5. Least or greatest number. "A is 14th from the front, B is 20th from the back, with 5 people between them" has two cases — A ahead of B (14 + 5 + 20 = 39) or B ahead of A (27). Draw both.
6. Comparisons. "P is taller than Q, R is shorter than Q …" — write one line with the tallest on the left and slot each person in as you read.

The single most common mistake is forgetting the − 1, or using it where it does not belong. Say what you are counting: the person themself, or only the people beside them.

How to check: rebuild the row from your answer — every given position must come out exactly.

Total = from the left + from the right − 1 · from the other end = total − position + 1 · people between = difference of positions from the same end − 1 · after an interchange, A's new place = B's old place · positions adding to more than the total have crossed · "k people between" has two cases; the least total is a + b − k − 2

In a row, Meera is 9th from the left and Asha is 14th from the right. They interchange places, and Meera becomes 17th from the left. How many people are in the row? → Meera's new seat is Asha's old seat, so that one seat is 17th from the left and 14th from the right: total = 17 + 14 − 1 = 30. Sanity check: Asha now sits 9th from the left, which is 30 − 9 + 1 = 22nd from the right — a real seat in a row of 30.