Order, Ranking & Comparison

At a glance 9 rules

1. 1Total from both endsIf X is $L$th from the left and $R$th from the right, the row holds $L + R - 1$ people: X is counted from both ends.
2. 2Rank from either endIn a row of $n$, the $k$th from one end is $(n - k + 1)$th from the other: 12th of 40 from the top is 29th from the bottom.
3. 3People between two ranksBring both ranks to the same end; then $p$th and $q$th have $|p - q| - 1$ people between. If ranks from opposite ends add up to more than $n$, the two have crossed.
4. 4Order unknown: two totalsX is $L$th from one end, Y is $R$th from the other, $k$ people between. Not crossed: $L + R + k$ people. Crossed: $L + R - k - 2$.
5. 5After a swapWhen A and B swap, A stands in B's old place. So A's new rank from one end plus B's old rank from the other end, minus 1, is the total.
6. 6Chain the comparisonsWrite each clue as bigger > smaller and merge them into one chain. Its ends are the extremes; anyone tied in by a single clue can still float.
7. 7'Only k above X''Only $k$ people above X' puts X in place $k + 1$: only two scored more than O, so O is 3rd. Anyone known to be below X cannot fill the places above X.
8. 8Signs along a chain$P \ge Q > R$ gives $P > R$: one strict sign makes the link strict. If the signs turn round, as in $P > Q < R$, P and R cannot be compared.
9. 9Add or subtract sumsWhen clues compare totals of weights or scores ($a + b = c + d$, $a + d > b + c$), add or subtract the two lines: shared terms cancel and leave one direct comparison.