Ranking and Ordering Sets in CAT DILR: Draw the Chain

DILR

Ranking and ordering sets look gentle. A handful of people, a few clues of the form A finished ahead of B, and questions about who came fifth. Yet these are the sets where aspirants build a neat table, start filling cells with guesses, and discover ten minutes later that two cells contradict each other.

The trouble is the table. A table has a slot for every position, so it invites you to place people before the clues have earned the placement. A chain does the opposite. It records only what the clues say, and lets the positions emerge. This guide shows how to draw the chain, and how to read positions off it.

Why the Table Misleads in Ranking Sets

A table is the right tool when each clue pins something to a specific cell, as in many arrangement sets. Ranking clues usually do something weaker. They say one person is ahead of another, without saying by how many places. A table has no good way to store that, so you end up writing tentative positions, and tentative positions are where errors breed.

The second problem is that tables hide structure. If A is ahead of B and B is ahead of C, the three of them form a line, and every later clue about any of them connects to that line. In a table, those connections are scattered across rows. In a chain, they are visible at a glance.

The idea is not new, and our guide on ranking-based DILR sets with relative ordering and unique positions covers the family of sets with solved examples. This post focuses on one method, the chain, and the position-counting trick that comes with it.

The Chain Draw: Four Steps

The method has four steps. The first three build the picture. The fourth reads positions from it.

Writing the higher-ranked person on the left means every arrow points the same way, so you never have to stop and think about direction. If a clue is phrased the other way round, as C finished behind B, you flip it as you write. Make that translation once, at the start, and your chain is consistent after that.

A Worked Set: From Clues to Chain

Here is a set written for practice. Six runners, A to F, finish a race with no ties. The clues are: A finished ahead of B. B finished ahead of C. D finished ahead of B. C finished ahead of F. E finished ahead of A.

Draw the arrows. From the first two clues, A then B then C. From the third, D sits ahead of B, though we do not know where relative to A. From the fourth, C is ahead of F, so the chain extends to F. From the fifth, E is ahead of A, so E joins at the front. Merging gives the main line E, A, B, C, F, with D branching in ahead of B.

Now count. Who is forced ahead of B? E, A and D, so three people. Who is forced behind B? C and F, so two people. That is five people accounted for, plus B himself makes six, which is the whole group. So B has exactly three above and two below, and B must finish fourth.

The same count gives C. Ahead of C are E, A, D and B, four people, and only F is behind. So C is fifth, and F is sixth. Positions one to three are shared among E, A and D, with E ahead of A. A question asking who finished fifth is answered at once: C. A question asking who finished last gets F.

Runner	Forced above	Forced below	Possible places
A	E	B, C, F	2 to 3
B	E, A, D	C, F	4 only
C	E, A, D, B	F	5 only
D	None	B, C, F	1 to 3
E	None	A, B, C, F	1 to 2
F	E, A, D, B, C	None	6 only

The Counting Rule: Best and Worst Positions

The table above rests on a simple rule. A person's best possible position is one plus the number of people who must be ahead of them. Their worst possible position is the group size minus the number of people who must be behind them. The range between those two numbers is where they can finish.

For A, one person is forced ahead, so the best place is second. Three are forced behind, so the worst place is six minus three, which is third. A can finish second or third. For D, nobody is forced ahead, so the best place is first, and three are forced behind, so the worst is third. D can finish first, second or third. These ranges match what a full enumeration would give.

When the best and worst positions coincide, the person is fixed. That is what happened with B, C and F. You did not have to try arrangements. The counts alone proved the positions. In CAT sets, questions about who must be in a particular place nearly always reduce to this check.

Count carefully when branches exist. A person is forced ahead only if there is a chain of clues leading from them to the person in question. Two people on different branches, such as A and D in the example, have no forced order between them. That freedom is exactly what the ranges express.

Answering the Usual Questions from the Chain

Ranking sets tend to ask a few kinds of questions, and the chain answers each. Who is in a given position? Check whether that position lies in a single person's range. Who could be first? List the people whose best position is one. How many orders are possible? Count the ways to arrange the unconstrained part.

In our example, who could finish first? D or E, since both have a best place of one. How many different finishing orders fit the clues? Positions four to six are fixed as B, C and F. Positions one to three are filled by D, E and A with E before A. There are three such orders: D, E, A; E, D, A; and E, A, D. So the answer is three, which you can confirm by listing.

Some questions add a new clue and ask what follows. Add it to the chain as a new arrow, then recount. This is where the chain earns its place over a table. A new clue is one more arrow, not a rebuild. Our guide on answering what-if questions without rebuilding the set describes the same idea for other set types.

Questions that say immediately ahead add a tight link. Two people joined by a tight link behave as a single block, and the block occupies two consecutive places. Treat the block as one unit when counting, and expand it afterwards. Many set-ups that look complicated simplify once you collapse such blocks.

When the Clues Are Not Simple Comparisons

Not every ordering clue is a bare comparison. You may meet clues such as, there are exactly two people between A and B, or C is not last. These do not fit as arrows, so record them next to the chain as side conditions and apply them after the basic structure is clear.

A clue with a fixed gap, like exactly two between A and B, tells you the positions differ by three. Once the chain narrows A and B to a few places, you can test which pairs are consistent. A negative clue, such as C is not last, removes a position from a range. Use it to trim the ranges you have already computed.

If a set mixes ordering with attributes, such as ages or heights, the same chain still works. The ranking is of the attribute, and the people are nodes. Keep the attribute explicit in your notes, so you do not mix up taller with older. Similar discipline is needed in our guide to seating arrangements and the four-step diagram lock, where the diagram plays the role the chain plays here.

Making the Chain Your Default Habit

You will not switch methods in the exam if you have not practised the new one. Take ordering sets from your practice list and redo them with the chain, even the ones you solved correctly with a table. Compare the time. Most aspirants find the chain is not slower once it is familiar, and it is far less error-prone.

Keep your notes tidy. Use a consistent direction, a consistent symbol for tight links, and a small table for ranges. Neatness is not decoration here. A clear chain lets you answer later questions in seconds, because the structure is already on the page.

Review the sets where you slipped. Was it a misread clue, a wrong arrow or a counting error? Each has a different fix. If you would like a plan for spreading DILR practice across your week, a CAT study planner can help, and you can talk to a CAT mentor if you want someone to watch how you work through a set.