Lesson 11.6 — Enumerate Cases and Test Feasibility
枚举情况与检验可行性
A strong case list is not random guessing. It defines every candidate, applies every clue, records every rejection, and proves what is possible, impossible, unique, or greatest.
Build a complete candidate ledger
Enumeration becomes proof only when the candidate set is complete and every survivor is checked against every condition.
The five-step habit
- Define the full candidate set.
- Apply the strongest clue first.
- Record why each candidate survives or fails.
- Construct an example when possibility or a maximum must be proved.
- Audit every original condition.
Warm-up candidate set
Select every integer from 1 through 12 that is even and not divisible by 3.
Selected: 0 candidates
| Candidate | Even? | Divisible by 3? | Survives both clues? |
|---|
Use every number card exactly once
This practice problem turns arithmetic clues into candidate pairs, then removes assignments that reuse a card.
Number-card puzzle
There is exactly one card for each whole number 1 through 9. Order within a two-card pair does not matter. Players A, B, C, and D each take two cards.
- A's two cards have sum 10.
- B's larger card minus the smaller card equals 1.
- C's two cards have product 24.
- The larger of D's two cards divided by the smaller equals 3.
Every card is used at most once. Which card remains?
Enumerate clue-specific pairs
Choose one pair for each player
Find an upper bound in the rotating-player game
A maximum needs a limit. Here the key limit is that a resting player must enter the next match and therefore cannot rest twice in a row.
Table-tennis rotation
A, B, and C play table tennis. Each match uses two players. Every match has a winner and a loser. The winner stays, and the loser is replaced by the resting third player in the next match. When A has played 9 matches and B has played 6 matches, what is the greatest possible number of matches C could have played?
Why rest counts matter
In every match exactly one player rests. If a player rests now, that player enters the next match. Therefore no player's rest marks may be adjacent.
For each tested total, compare A's, B's, and C's required rest counts with that maximum.
| Total matches M | C appearances | A rests | B rests | C rests | Max rests for one player | Feasible? |
|---|
Construct the maximum case
An upper bound is not enough. We must exhibit an actual match schedule that reaches it.
A 13-match rest schedule
The row records who rests in each match. Adjacent resters are different, so the schedule can occur: the next rester is the current match's loser.
Enumerate periodic alignments
Two repeating reports create a 12-position combined cycle, but the second report's alignment depends on the line length.
Soldiers reporting from opposite ends
Soldiers stand in one line. First they report from left to right using the repeating pattern 1, 2, 3, 4, with the leftmost soldier starting at 1. Then they report from right to left using the repeating pattern 1, 2, 3, 4, 5, 6, with the rightmost soldier starting at 1. Exactly five soldiers report 3 both times. What is the greatest possible number of soldiers?
Systematic scan
Why 12 positions, and why can the scan stop?
At position i from the left, the first report is 3 when i = 3, 7, 11, … . The second report is 3 when N − i leaves remainder 2 on division by 6. Both conditions repeat after 12 positions because 12 is divisible by both 4 and 6.
If N is even, the required positions have different parities and cannot meet. For odd N, this table identifies the first double-3 position; add 12 repeatedly.
| Remainder of N ÷ 12 | First matching position | First N in this class with at least six matches |
|---|---|---|
| 1 | 11 | 73 |
| 3 | 7 | 75 |
| 5 | 3 | 65 |
| 7 | 11 | 79 |
| 9 | 7 | 69 |
| 11 | 3 | 71 |
Within each remainder class, increasing N by 12 adds one matching position. Every odd N above 67 is therefore already at or beyond that class’s six-match threshold. The only five-match lengths are 53, 57, 59, 61, 63, and 67.
If the line length is even, the two kinds of “3” fall on opposite parities, so there are no double-3 reporters. For an odd length, double-3 reporters repeat every 12 positions.
Scanning through 80 is enough: every odd length greater than 67 already contains a sixth double-3 position, while every even length contains none. Thus no later length can return to exactly five.
Enumerate distinct shooting-score sets
The score total and best-shot limit reduce a large-looking problem to four candidate sets.
Shooting practice
A, B, and C each fire four shots, and every shot hits. Each shot scores a positive whole number from 1 through 7 rings. Each person's four scores are different and total 17. B has exactly two score values in common with A. A and C have exactly one score value in common. What is that common score?
Step 1 · Generate every four-number set
Choose four distinct positive integers from 1 through 7 whose sum is 17. Order does not matter.
Step 2 · Test an assignment
Choose three candidate sets.
Step 3 · Look for an invariant
More than one full assignment survives. The question is still answerable because the one score shared by A and C is the same in every valid assignment.
List possible and impossible scores
A score is feasible exactly when it can be written as 6 × wins + 5 × draws. Losses add zero and do not change the score.
Game-score problem
Each round gives 6 points for a win, 5 for a draw, and 0 for a loss. The number of rounds is unrestricted. How many positive whole-number scores can never be obtained?
Choose a score
Click a score to see whether a nonnegative combination of 6-point wins and 5-point draws exists.
Why the search can stop
The five consecutive scores 20, 21, 22, 23, and 24 are all attainable:
Adding another 5-point draw produces the next block of five scores. Therefore every score at least 20 is attainable.
Name exactly what the evidence proves
Different case analyses establish different conclusions. Do not overclaim.
Independent workshop
Solve at least 6 of 8 fresh problems. Each has a local hint.
Hint
List multiples of three, then remove the even ones.
Hint
List smaller card first so no pair is counted twice.
Hint
Try each possible number of seven-point rounds.
Hint
Find four consecutive attainable totals. Explain why adding another four-point round then covers every larger total.
Hint
Compare with the largest possible four-number sum.
Hint
A player participates whenever they are not resting.
Hint
The resting person must enter the next match.
Hint
Distinguish a fixed requested value from a unique full case.
Independent exit ticket
Solve all five fresh problems. The certificate also requires Missions 1–9.
Hint
List every multiple of four in the range, then filter.
Hint
Start with the smallest possible smaller card.
Hint
Test small totals and find three consecutive attainable ones.
Hint
A consecutive block must cover every remainder when adding three.
Hint
Count the matches in which C is not the resting player.
Optional reflection — not automatically graded
Certificate of mastery
Case Enumeration & Feasibility Architect
This certifies that the learner can define complete candidate sets, eliminate impossible cases, construct attainable extremes, use periodicity, and state exactly what a case analysis proves.
Lesson 11.6 · Grade 5 Mathematical Reasoning
Teaching notes
The warm-up ledger, exhaustive-search displays, explicit constructions, general labels, workshop, and exit ticket are added instructional scaffolds.