DILR Game and Quiz Sets: Back Out the Scoring Rules

DILR

Some DILR sets give you the results of a quiz or a tournament, but not the rules. You are told that a player scored 22 points, another scored 16, and you must work out how many points a correct answer is worth or what a draw earns. The rules are in the totals, and getting them out is the first job.

This guide shows a clean way to do it. You set up one equation per player, solve for the rules, and then use the rules to answer the questions. Along the way, you will see how small integer constraints often fix the answer even when the equations alone seem to leave room. Every example below was solved and checked by substitution.

What These Sets Look Like

A quiz set may list, for each of several players, the number of questions answered correctly, answered wrongly and left blank, along with a final score. A tournament set may list wins, draws and losses along with points. In both cases, the scoring rule is a fixed number of points per result type, and it is hidden.

The set may announce the rules, in which case the work is easy. When it does not, it usually gives enough rows to let you solve for them. Some sets give only partial information, such as the total score and the number of matches, and ask you to find possible combinations. Those are puzzle style and need constraints.

Whichever the form, the starting step is the same. Decide what the unknown is, usually the points for each result type, and write one equation for each row that gives a total.

Setting Up the Equations

Let a be the points for a correct answer and b the penalty for a wrong one. Suppose Player A answered 6 questions correctly and 2 wrongly, and scored 22. Player B answered 5 correctly and 4 wrongly, and scored 16. Then 6a minus 2b equals 22, and 5a minus 4b equals 16.

Solve the pair. Multiply the first equation by 2 to get 12a minus 4b equals 44. Subtract the second equation, 5a minus 4b equals 16, and you get 7a equals 28, so a equals 4. Put a equals 4 back into the first equation: 24 minus 2b equals 22, so b equals 1. A correct answer scores 4 and a wrong one costs 1.

Check with both rows. Player A: 6 times 4 is 24, minus 2 is 22. Player B: 5 times 4 is 20, minus 4 is 16. Both match. A third player, with 7 correct and 1 wrong, would score 28 minus 1, which is 27. Once you have the rules, every later question is arithmetic.

Player	Correct	Wrong	Total score	Equation
A	6	2	22	6a minus 2b equals 22
B	5	4	16	5a minus 4b equals 16
Result	a equals 4	b equals 1	Check	24 minus 2 equals 22 and 20 minus 4 equals 16

Using Small Integer Constraints

Sometimes there are fewer equations than unknowns. That is where integer constraints help. In a tournament, the number of matches a team plays is fixed, and wins, draws and losses must be whole numbers that add up to it.

Take a team that played 5 matches and scored 11 points, with 3 points for a win and 1 for a draw. Let w be wins and d be draws. Then 3w plus d equals 11, and w plus d is at most 5. If w is 3, d is 2, and w plus d is 5, which fits. If w is 2, d would be 5, and w plus d would be 7, which is too many. If w is 4, d would be negative. So the only possibility is 3 wins, 2 draws and no losses.

This kind of reasoning appears in many sets. You list the possible values of one variable, compute the other, and test against the constraint. With small numbers, the list is short, and you often end with one answer.

Tournament Sets With Points Tables

In a round robin tournament, every team plays every other team once. With n teams, the number of matches is n times n minus 1, divided by 2. The sum of the points of all teams equals the points awarded per match times the number of matches, as long as each match gives the same total. If a win gives 3 and a draw gives 1 each, a drawn match gives 2 and a decisive match gives 3, so the total depends on the number of draws.

That observation is useful. If the table shows the total points of all teams, you can find the number of drawn matches. Suppose 6 matches were played, with a total of 16 points across all teams. If every match was decisive, the total would be 18. Each draw reduces the total by 1, so there were 2 draws.

Rules such as a win giving 2 points and a draw giving 1 each give a fixed total of 2 per match, and the total points tell you nothing about the number of draws. Check which rule applies before you use the total.

Quiz Sets With Rounds and Bonuses

Some quiz sets add rounds with different values, or a bonus for a streak. When rounds have different values, treat each round as its own set of unknowns, but look for patterns. A common design is that later rounds are worth a multiple of the first. Test this idea early, because it cuts the number of unknowns.

If a bonus applies only in some cases, write it as a separate unknown with a note of when it applies. Rows where the bonus did not apply give you the base rules directly. Then use the other rows to find the bonus. This is a way of decoupling the unknowns.

The check is the same as before. Use all the rows. A rule that fits only some rows is wrong, however neat it looks. Our guide on missing data tables has a similar approach to filling blanks from totals, and the habits carry over.

Reading the Set Before You Solve

Spend the first thirty seconds reading what the set does not say. Are blank answers allowed, and do they score zero? Is there a fixed number of questions per player, so that correct, wrong and blank always add up to the same total? Is the penalty applied to every wrong answer, or only after a certain number? Each of these facts is an extra equation or constraint, and the set often hides it in a single sentence.

Take the quiz above. If every player answered all 10 questions, then correct plus wrong plus blank equals 10 for each row, and Player A with 6 correct and 2 wrong must have left 2 blank. That detail is useful when a question later asks how many questions a player left blank, since you can read it from the row without any new calculation.

Also look for rows that are partly missing. A table may give the score and the number of correct answers, but leave out the wrong answers. With the rules known, you can fill the gap: if a player scored 18 with 5 correct answers and a equals 4, then 20 minus the penalty total equals 18, so there were 2 wrong answers. Missing cells are the reason to extract the rules first.

Answering the Questions After the Rules

Once the rules are fixed, the questions are usually simple: the maximum possible score for a player, the minimum number of correct answers to reach a target, or the rank of a player given certain results. Convert each into arithmetic using the rules.

For maximum and minimum questions, set the other values to their extremes. If a player must answer at least some questions to qualify, the best score assumes all remaining answers are correct, and the worst assumes all are wrong or blank. Remember that blank answers may carry no penalty, which changes the minimum.

Keep a clean table of rules and results. A set can have five or six questions that all use the same rules, and a clean table makes each question quick. This is the benefit of doing the extraction carefully at the start.

If you want to see how sets fit into your plan, use the CAT study planner, and if you want feedback on your set selection, you can talk to a CAT mentor.