Casework compass
Every valid result belongs to exactly one case.
Add across cases because the cases are alternatives. Multiply within one case when several choices are made independently.
Total = Case 1 + Case 2 + ···
Chapter 24 · Classification
Build Complete, Non-Overlapping Case Tables
When there are many possibilities, do not chase them randomly. Choose one control feature, give every result exactly one case label, count each case, and then add the case totals.
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Foundation
Recognize a valid classification
Your case-table routine
- State what counts as one outcome. Does order matter? May an item repeat?
- Choose one feature that assigns every outcome to exactly one row.
- Count within each row; then add the rows.
- Check one example in every row, including the smallest and largest cases.
Multiply choices made together; add cases that cannot happen together. Predict before opening a complete list.
A useful case system must be complete and non-overlapping. Choose the only valid classification of the numbers 1 through 12.
Flawed system A audit
Correct operation language
Checkpoint
Choose the valid system above, then choose the two correct operations.
Need a hint?
Try putting 6 into each proposed system. It must fit exactly one case.
Worked example 1
Classify subsets by their total
A subset uses each number at most once. Changing its order makes no new subset. This task excludes the empty subset. Its sum is 0, which is divisible by both 3 and 5, but an empty selection is not an allowed outcome.
Choose one or more numbers from {1,2,3,4,5,6}. The sum must be divisible by 3 but not divisible by 5. The worked example classifies each valid subset by its sum.
Explore one subset
0current sum
nodivisible by 3
nodivisible by 5
noqualifies
A subset belongs to exactly one sum case, so this case label cannot overlap.
original totals
6sum cases
19valid subsets
63nonempty subsets checked
0duplicates
The valid sum cases are 3,6,9,12,18,21. Sum 15 is excluded because it is divisible by 5.
| Case label: subset sum | Subsets in this case | Count |
|---|
Checkpoint
Need a hint?
List the possible sums divisible by 3, then remove sums divisible by 5. Classify each nonempty subset by its sum.
General laboratory
Generate a complete subset-sum case table
63subsets checked
6sum cases
19qualifying subsets
5largest case
| Sum case | Count | Sample subsets |
|---|
Checkpoint challenge
Run the preset 1–7, divisible by 4, but not divisible by 3.
Need a hint?
Keep the same rules when changing the pool: no repeats, no order, and no empty subset.
original target puzzle
Classify by multiplicities, not shot order
Build five hits totaling 100, then check your construction. After an attempt, the separate worked review explains why the solution is unique. A different order of the same hits is the same case.
A player fires exactly five shots for a total of 100 points. The ring values are 47,36,27,9,5. Because shot order does not matter, classify by how many times each score occurs.
0shots used
0points
100distance from 100
not readystatus
Complete search. The complete search checks every nonnegative multiplicity vector whose entries total five. It finds 1 valid case.
Checkpoint
Need a hint?
Count by how many times each score occurs. Both the number of hits and their total must match.
Test 24 transfer
Count a round-robin tournament without double-counting games
One case label per game
Each pair of different players plays exactly one game. Classify every game by the lower-numbered player. Player 1 contributes games against all later players; Player 2 contributes only the games not already counted, and so on.
9+8+···+1=45
Case table
9games in first case
1games in last case
45total games
45unordered pairs
Checkpoint for exactly 10 players (regardless of the laboratory setting)
Need a hint?
Give each match to the lower-numbered player. That assigns it exactly once.
Test 24 transfer
Distribute identical items by fixing the first group
Eight identical machines are distributed among three labeled workshops A, B, and C. Every workshop receives at least one machine. Classify by the number sent to A.
Case table
Try one distribution
6+5+4+3+2+1=21
6A-cases
6ways when A=1
1ways when A=6
21total distributions
Checkpoint for exactly 8 machines (regardless of the laboratory setting)
Need a hint?
Fix the first choice, then list every compatible second choice. Check the last row too.
exercise
Use parity cases and a minimum-total bound
Fifteen different positive integers have total 123. What is the greatest possible number of odd integers?
Candidate odd counts
A construction that reaches the bound
Use the nine smallest positive odd integers and the six smallest positive even integers:
81+42=123
The number of odd addends must itself be odd because the total 123 is odd.
15different integers
123required total
9greatest feasible odd count
123minimum total for 9 odds
Checkpoint
Need a hint?
The number of odd terms must be odd because the total is odd. For each candidate count, add the smallest distinct odd and even terms and compare with 123.
Case-table audit laboratory
Detect overlap and build a clean remainder partition
First task: count masses with balance weights
Use weights 2 g, 5 g, and 20 g, each at most once. In this activity, all weights go on the pan opposite the object. Count positive masses only; order does not matter.
Hint: classify by the largest weight used
Largest 2: one case. Largest 5: use 5 alone or add 2. Largest 20: decide independently whether to include 2 and whether to include 5.
Compare your complete list
1 + 2 + 4 = 7 masses: 2, 5, 7, 20, 22, 25, 27 g. There is no empty case.
If weights were allowed on both pans, subtraction would be possible too. That is a different problem, with 13 positive masses: 2, 3, 5, 7, 13, 15, 17, 18, 20, 22, 23, 25, 27 g.
Flawed categories on 1 through 12
Cases: even, multiple of 3, neither. Every number is covered, but multiples of 6 are counted twice.
2overlap values
0omitted values
Clean remainder partition
Every integer has exactly one remainder, so remainder classes are automatically complete and disjoint.
Checkpoint
Need a hint?
List cases by their decisive feature; check the smallest and largest possibilities.
Independent practice
Case-table workshop
Correct all eight questions to complete the workshop.
1. A valid case system is complete and…
2. Choose a nonempty subset of {1,2,3,4,5,6}, with sum divisible by 3 but not 5. How many possible sum values qualify?
3. How many subsets satisfy the conditions in Question 2?
4. Five hits score 100 using scores 47,36,27,9,5. How many hits score 27?
5. How many games are in a 10-player round robin?
6. How many positive distributions of 8 identical machines among A, B, C?
7. Fifteen different positive integers total 123. What is the greatest possible number of odd integers?
8. For 1–7, sum divisible by 4 but not 3: qualifying subsets?
Need a hint?
Answers: non-overlapping; 6; 19; 2; 45; 21; 9; 21. The subset table groups by sum; the five-shot solution is 36+27+27+5+5; tournament games total 9+8+…+1; machine distributions total 6+5+…+1. For the odd-count question, nine odds and six evens have minimum sum 123.
Objective assessment
Exit ticket and certificate
Earn 5 out of 5. The certificate also requires Missions 1–9 to be complete.
1. Across alternative cases, we usually…
2. How many subsets of {1,2,3,4,5,6} have sum 12? Each number is used at most once; order does not matter.
3. Five hits total 100, each scoring 47,36,27,9 or 5. How many different score-count combinations work? Ignore shot order.
4. A 12-player round robin contains how many games?
5. Distribute 10 identical items among 3 labeled nonempty groups. How many ways?
Need a hint?
Answers: add; 5; 1; 66; 36. Sum 12 has five subsets. The five-shot puzzle has one multiplicity case. For 12 players, 12×11÷2=66 games. For 10 items, fix A=1,…,8; the positive B–C counts are 8+7+…+1=36.
Chapter 24 · Classification
Complete Case-Table Architect
This certifies that the learner can build case systems that are complete, non-overlapping, and correctly counted.
Lesson 24.1