CAT DILR What If Questions: Answer Without a Rebuild

DILR

Question three of the set reads: "If the third condition is replaced by the following, what is the new arrangement?" Half the room reaches for a fresh page and starts again from zero. Four minutes later, the redrawn set is mostly identical to the one already sitting on the old page.

Rebuilding feels safe because it is thorough. It is also slow, and it throws away the work you already did. There is a faster way, and it relies on one habit you can build while solving the base set: recording why each deduction was made. This guide shows how to answer "what if" questions without redrawing the world.

Why What If Questions Make People Redraw Everything

What if questions are common in DILR sets across arrangements, games and scheduling, though set types and counts change from year to year. They reward the aspirant who understands how the set hangs together, because the change touches a few cells and leaves the rest alone.

The instinct to redraw comes from not knowing which cells depend on which condition. If you finished the base set by a mix of forced moves and hunches, you cannot tell afterwards which parts were solid. So you start again, which feels safer than guessing which part to trust.

The fix is to make the dependency visible while you solve. When you force a cell, jot the number of the condition that forced it. That costs a second per deduction. It saves minutes later, because a what if question becomes a short lookup: which deductions mention the condition that changed?

This also makes the base set more reliable. If you notice that a deduction cites no condition at all, it is a guess, and you should treat it with suspicion. That small discipline catches wrong starts earlier, which is a topic we cover in DILR sets requiring unstated assumptions.

The Delta Overlay: Four Moves for Any What If

The Delta Overlay is a short routine. It treats the what if as a small change layered over the base, not as a new puzzle.

A quick word on the order of the moves. Freezing the base comes first because everything else compares against it. Overlaying the change second forces you to say precisely what changed, which catches misreadings of the stem early. Rippling third is the only step that needs real reasoning. Restoring last is a habit more than a step, but it is the one most often forgotten.

The word delta means the difference. You are not asking what the whole set looks like now. You are asking what changes. Often the answer is small: two people swap places, or a single value shifts, and everything else stays. Knowing that in advance lets you check your answer against the base for plausibility.

The ripple step is where most of the thinking happens. Once you reopen a deduction, new forced moves may follow, and each one may reopen others. Keep going until no more forced moves remain. If the ripple stops quickly, you are done. If it spreads across the whole diagram, the changed condition was central, and a rebuild may genuinely be faster.

The Dependency Ledger in a Worked Base Set

Here is a small set written for practice. Five people, P, Q, R, S and T, stand in a queue in positions one to five. Condition one: Q stands immediately behind P. Condition two: R is in position three. Condition three: T is not at either end. Condition four: S is ahead of P.

Solve it with a ledger. R is in three by condition two. P and Q are adjacent with Q after P, so the pair sits in positions one and two, or four and five, since position three is taken. If the pair is in one and two, S cannot be ahead of P, so condition four rules it out. The pair is in four and five, citing conditions one, two and four.

That leaves positions one and two for S and T. Condition three says T is not at an end, so T is in two and S is in one, citing condition three. The base arrangement is S, T, R, P, Q, and the ledger now says which conditions each position rests on.

Notice the pattern in the ledger. The positions of P and Q rest on conditions one, two and four. The positions of S and T rest on condition three. That division is what makes the what if questions quick.

Why bother with the ledger when a set has only four conditions? Because the sets in the exam have more, and the same discipline scales. With eight conditions and a dozen forced cells, a what if that touches one condition may reopen only two cells, and you will find them in seconds if the labels exist. Without labels you would reread the whole set to rediscover which cells mattered.

The ledger also protects you from a subtle error: reusing a deduction that quietly depended on the condition that changed. That is how a what if produces an answer that is almost right, with one person in the wrong place. The label shows the dependency, and the dependency shows what to recheck.

Two What Ifs on the Same Set

Suppose the first what if replaces condition three with "T is at one of the ends". Check the ledger. The positions of P and Q do not cite condition three, so they stay at four and five. Only the placement of S and T reopens. With T at an end and positions one and two available, T takes one and S takes two. The new arrangement is T, S, R, P, Q. Only two people moved.

Now suppose the second what if replaces condition four with "S is behind Q".

Condition four sits under the position of P and Q, so that deduction reopens. If the pair is in four and five, S would have to be in position six, which does not exist. So the pair is in one and two. S and T take four and five, and T is not at an end by condition three, so T is in four and S is in five. The new arrangement is P, Q, R, T, S.

Compare the two. The first what if touched condition three and left the pair alone. The second touched condition four and reordered almost everything. The ledger told you in advance how big each change would be. That prediction is the real value of recording dependencies.

Do one last consistency check before you commit to a what if answer. Read every condition, original and changed, against your new arrangement. It takes under half a minute and it catches the rare slip where a reopened cell has been left in its old state. A what if answer that satisfies all the current conditions is almost certainly right, because arrangement sets rarely have more than one valid solution once the conditions are complete.

Three Kinds of What If and What Each Reopens

Not every what if is a replacement. Name the kind first, because each reopens a different part of the set.

Kind of what if	What it does	What to reopen
Replace a condition	Swaps one rule for another	Cells that cited the old rule
Remove a condition	Drops a rule entirely	Cells that cited it, now with more options
Add a condition	Imposes an extra rule	Prune the surviving cases against the new rule

Removing a condition usually widens options, so the answer may become a range or a set of cases. Adding a condition does the opposite and narrows them. If your base set had multiple valid cases, keep a case tree beside the diagram. An added condition then prunes branches, and the answer is whatever survives.

When a what if asks for something that holds in every case, check whether your base already has an answer that is common to all branches. Some questions can be answered from the fixed skeleton without drawing anything new. That idea appears in our guide to DILR questions solvable without a full arrangement.

A what if that adds a condition to a set with a unique base solution deserves a quick look. If the added condition is already satisfied by the base, the answer is unchanged. If it contradicts the base, then the question is really a replacement in disguise, and you should reopen the cells it conflicts with. Check the added rule against the base before doing anything else.

A Practice Routine for What If Questions

Practise the ledger first, because the rest follows from it. For the next two weeks, annotate every forced deduction with its condition number on every set you solve. It feels slow at first, and speed returns once the habit settles. After the base set, run each what if through the four moves.

After each set, ask which what if took longest and why. Usually it is one of three causes: you forgot to record a dependency, you overwrote the base, or the change was central and a rebuild was reasonable. Naming the cause tells you what to fix.

If a set looks impossible at first and gets easier on a second read, a different mindset applies, as described in sets that look impossible on first read and easy on the second. For extra material when you run out of official sets, see how to practise DILR when official sets run out.

Games and tournament sets often carry what if questions too, where a single changed result alters a table. Try CAT DILR games and tournament sets after arrangements, and apply the same ledger. For a view of how real papers phrase them, use our topic wise CAT previous year questions.