DILR
Venn Diagrams
A one-page CAT 2026 reference for Venn diagrams and set theory, covering the two-set and three-set union formulas, only-one and neither regions, exactly-two and exactly-one breakdowns, the full…
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Games & Tournaments
CAT’26 DILR CHEATSHEET
Every match-count, points table and tie-break rule you need for CAT 2026.
Games and tournaments questions read like sports trivia but score like pure logic, and the gap between the two is where most of the marks are lost. A knockout draw, a league points table, or a mixed group-stage-then-knockout format each carry their own match-count formula, and a points table almost always hides a reconciliation the question expects you to spot without being told. This sheet lays out every standard setup, from basic knockout and round-robin counts, through points scoring, qualification ceilings and tie-break order, to reading match results backward out of a finished table. Every box carries one worked example with real numbers, so you drill the counting and the reading together, not just the formula on its own.
Every games and tournaments rule you need
1Knockout Matches to Find a Winner
A single-elimination draw needs exactly one loss per eliminated team.
Matches in a knockout of n teams = n − 1
Example. A 32-team knockout tournament needs 32 − 1 = 31 matches to produce one champion.
CAT Hack. Every match removes exactly one team, so the match count is always teams minus the single winner left standing.
2Knockout Rounds and Byes
When the team count is not a power of 2, some teams skip round one.
Byes in round 1 = next power of 2 above n, minus n
Example. For 20 teams, the next power of 2 is 32, so 32 − 20 = 12 teams get a first-round bye.
CAT Insight. A bye still counts as advancing a round, but it is not a match, so do not add it to the match total.
3Round-Robin (League) Total Matches
Every team plays every other team exactly once.
Matches in a league of n teams = nC2 = n(n−1)/2
Example. An 8-team league has 8 × 7 / 2 = 28 matches in total, one between each pair.
CAT Favourite. This nC2 count is the single most tested number in tournament DILR sets, so keep it ready without recalculating.
4Double Round-Robin
Every team plays every other team twice, typically home and away.
Matches in a double league of n teams = n(n−1)
Example. A 6-team double round robin has 6 × 5 = 30 matches, twice the single round-robin count.
CAT Hack. Just double the single round-robin count, n(n−1)/2, rather than rederiving it from scratch.
5Points Table Reconciliation
Games played must equal the sum of wins, losses and draws for each team.
Played = Wins + Losses + Draws
Example. A team with 10 played, 6 wins and 2 draws has 10 − 6 − 2 = 2 losses, filling the row exactly.
CAT Favourite. Whenever one column of a points table is missing, this reconciliation almost always fills it in directly.
6Standard Points Scoring
Most league tables award fixed points per win, draw and loss.
Points = (Wins × win value) + (Draws × draw value)
Example. With 3 points a win and 1 a draw, a team with 5 wins and 4 draws has 5(3) + 4(1) = 15 + 4 = 19 points.
Common Mistake. Always confirm the win and draw values stated in the question. CAT sets do not always use football's 3-1-0 system.
7Maximum Possible Points
The ceiling a team could still reach if it wins every remaining game.
Max points = current points + (remaining games × win value)
Example. A team on 14 points with 4 games left, each worth 3 for a win, can reach at most 14 + 12 = 26 points.
CAT Hack. Use this ceiling to instantly rule out teams that cannot catch the leader even with a perfect run.
8Minimum Matches to Guarantee Qualification
The smallest win count that clinches a spot regardless of other results.
Guaranteed if max points of rivals for the last spot < your current points
Example. If no rival can exceed 18 points even with all wins left, and you already have 19, qualification is already guaranteed.
CAT Insight. Qualification is not about your own remaining games alone. Check every rival's own maximum ceiling too.
9Group Stage Plus Knockout
Teams first play a round-robin within groups, then a knockout follows.
Total = (group stage matches per group × groups) + knockout matches
Example. 4 groups of 4 in round robin give 4 × 6 = 24 group matches, plus a knockout of the 4 group winners adds 3 more, so 27 matches overall.
CAT Favourite. Split this hybrid format into its two pieces and total each with its own formula, group stage and knockout separately.
10Seeding and Bracket Position
Higher-seeded teams are placed to avoid meeting early in the draw.
Top 2 seeds are placed in opposite halves of the bracket
Example. In a 16-team seeded draw, seed 1 and seed 2 can only meet in the final, since they anchor opposite halves of the bracket.
CAT Insight. Seeding clues usually fix which half of the draw a team sits in, which narrows down every match it could possibly play.
11Tie-Breaking Rules
When two teams finish level on points, a stated rule decides the order.
Apply tie rules in the order given, such as head-to-head then net score
Example. Two teams level on points are separated by checking who won their head-to-head match first, and only if that also ties do you move to the next rule.
Common Mistake. Never skip straight to net run rate or goal difference. Apply the tie-break rules strictly in the order the question states.
12Deducing Match Results from a Table
Working backward from a final points table to individual match outcomes.
Cross-check each team's wins, draws and losses against every rival
Example. If team A's only draw must be against the one other team that also shows exactly one draw, that pairing is the fixed match between them.
CAT Hack. Start from the team with the fewest games left to explain. It usually has the fewest possible combinations to check.
13Matches Needed to Rank Everyone
Ranking every player from a set of head-to-head matches, not just finding a winner.
Full ranking still needs more than n − 1 matches from a single knockout
Example. A straight knockout only ranks the winner reliably. Ranking all 8 players in strict order typically needs extra playoff matches beyond the 7 knockout matches alone.
CAT Insight. A pure knockout tells you the champion, not the full order. Extra matches, like a losers' bracket, are needed to rank everyone else.
Exam shortcuts, common traps and revision
14
CAT Exam Shortcuts
- Knockout matches for n teams = n − 1
- Round-robin matches for n teams = n(n−1)/2
- Double round-robin = n(n−1)
- Played = wins + losses + draws, always
- Max possible points = current points + remaining games × win value
- Split hybrid formats into group stage and knockout, then add
- Apply tie-break rules strictly in the stated order
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Most Common CAT Traps
- Adding a bye into the match count instead of leaving it as a free pass
- Assuming a football-style 3-1-0 points system without checking the question
- Ruling a team out of qualification without checking every rival's own ceiling
- Jumping to net run rate or goal difference before checking head-to-head first
- Forgetting a knockout only ranks the champion, not everyone else
- Miscounting a double round robin as the single round-robin value
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30-Second Revision Box
- Knockout = n − 1 matches, league = n(n−1)/2 matches
- Played always equals wins plus losses plus draws
- Max points uses every remaining game as a win
- Check every rival's ceiling before confirming qualification
- Tie-breaks apply strictly in the order given
- A knockout finds a champion, not a full ranking
Most tournament questions fall apart the moment you separate the format from the table, so name the format first, whether it is knockout, league, or a hybrid of the two, and only then start filling in numbers. The reconciliation rule, played equals wins plus losses plus draws, quietly solves more of these questions than any other single fact. When you want an honest read on where you stand, run a full mock and check your projected percentile with the Optima Learn CAT score predictor, then work through the DILR breakdowns on the blog. If you would rather build a plan with a person, book a free call with an Optima Learn mentor and shape one to your target.
