Order and ranking problems involve determining relative positions, heights, scores, ages, or ranks of individuals or objects arranged in a sequence or comparison chain. Questions test your ability to calculate total count, individual positions, positions after interchanging, and persons sitting between two points.

Single row ranking formulas

When the position of a single person is given from both ends (left/top and right/bottom):

• Total number of persons ($T$): $T = L + R - 1$, where $L$ is rank from left/top and $R$ is rank from right/bottom. Subtracting 1 corrects double-counting the person.
• Rank from Left ($L$): $L = T - R + 1$.
• Rank from Right ($R$): $R = T - L + 1$.

Position interchanging

When two individuals swap positions in a row:

• Calculate total strength using the new rank of the person who moved and the original rank of the other person from the opposite end: $T = L_A' + R_B - 1$.
• Number of persons between them equals the absolute difference between initial and new positions of the person who moved minus 1: Persons between = $|L_A' - L_A| - 1$.
• The second person shifts by the exact same number of places in the opposite direction.

Persons sitting between two individuals

To find the number of persons between person A (rank $L$ from left) and person B (rank $R$ from right) in a row of total $T$ persons:

• Non-overlapping case ($T \ge L + R$): Persons between = $T - (L + R)$.
• Overlapping case ($T < L + R$): Persons between = $(L + R) - T - 2$. Subtracting 2 excludes both boundary individuals.

Maximum and minimum capacity of a row

Given rank of A from left ($L$), rank of B from right ($R$), and $N$ persons between them:

• Maximum total ($T_{max}$): $T_{max} = L + R + N$ (Non-overlapping condition).
• Minimum total ($T_{min}$): $T_{min} = L + R - N - 2$ (Overlapping condition, valid when minimum of $L, R \ge N + 2$).

Order and comparison puzzles

For height, weight, age, or score comparisons:

• Use inequality signs ($>$, $<$) to represent each statement.
• Chain multiple statements together (e.g., $A > B > C$).
• Maintain separate tracks if relations are ambiguous until all clues are combined.

A reliable solving method

1. Identify whether the problem involves position ranking or inequality comparison.
2. For row ranking, write down given parameters ($T$, $L$, $R$, or $N$).
3. Check if positions overlap by comparing $(L + R)$ with $T$.
4. Apply the corresponding formula ($T = L + R - 1$ or overlapping adjustment).
5. For comparison problems, draw a linear order chain from highest to lowest.

Common mistakes

• Forgetting to subtract 1 when deriving total strength from two end ranks.
• Using non-overlapping formula when the positions overlap ($T < L + R$).
• Forgetting to subtract 2 for overlapping cases when counting persons between.
• Assuming fixed absolute positions when only relative inequalities are given.

Quick revision summary

• $T = L + R - 1$ for single person rank from both ends.
• Non-overlapping between = $T - (L + R)$.
• Overlapping between = $(L + R) - T - 2$.
• Maximum total = $L + R + N$; Minimum total = $L + R - N - 2$.