Capacity-value backpack
Choose whole items by total value while checking a shared capacity.
Adult planning suggestion: about9–12 · Logic & rule checking
- Players
- 2–3 per group
- Time
- 15–20 minutes
- Materials
- Four item cards and capacity 7 mat; no real load.
Prepare
Items are fictional: A weight 2/value 3, B3/4, C4/7, D5/8. The backpack capacity is 7. There is exactly one copy of each item. A choice is a subset, not a sequence; taking A then C and C then A is the same selection.
Play
Choose any whole items whose total weight is at most 7 and add their values. You may leave unused capacity; you may not split a card, reuse an item or confuse value with weight. Partners propose a selection and another checks its two totals before comparing it with alternatives.
Worked example
A+C weighs 6 and values 10. B+C weighs 7 and values 11. A+D also weighs 7 and values 11. C+D has value 15 but weighs 9 and is illegal. Choosing only the single highest-value item D gives 8, which does not prove an optimal subset.
Finish and check
There are sixteen subsets including empty. Enumerate them or systematically rule out overweight combinations. The maximum here is 11, with two distinct selections A+D and B+C. Accept both; filling capacity is not the goal unless it also gives the highest value. Finish with one optimal selection and a reason no legal alternative beats it.
Adaptation and limits
A class of thirty can use ten trios with selector, weight checker and value checker. Numbers are abstract marks rather than a prescription for children to carry loads. The technical reference defines the 0–1 choice, not evidence about this age group. For another capacity, create a new answer rather than reusing 11.
Purpose, observation and provenance
Within this activity: Choose whole items by total value while checking a shared capacity.
Ask whether an attractive value is feasible and whether an alternative optimum was missed.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
CP Algorithms — Knapsack problem — Technical whole-item capacity/value definition; our four-item instance and paper task are original.
Age bands are planning suggestions. Goals describe observable actions, not measured long-term gains or institutional endorsement.
Free alternative: Copy the supplied grids, labels and cards onto reused paper. No specialist product is needed. Adults may prepare pieces or act as a scribe.
Printable materials
Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.
Player materials · Capacity-value backpack
A(2,3), B(3,4), C(4,7), D(5,8); capacity 7.
Subset / weight sum / value sum / legal?
Capacity7. One copy each.
Player record · keep host key separate
| Subset | Weight | Value | Legal? |
|---|---|---|---|
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
Host key · separate from players
Best 11 from A+D or B+C. C+D is overweight 9. Sixteen subsets checked.
Run the round
Items are fictional: A weight 2/value 3, B3/4, C4/7, D5/8. The backpack capacity is 7. There is exactly one copy of each item. A choice is a subset, not a sequence; taking A then C and C then A is the same selection.
Choose any whole items whose total weight is at most 7 and add their values. You may leave unused capacity; you may not split a card, reuse an item or confuse value with weight. Partners propose a selection and another checks its two totals before comparing it with alternatives.
A+C weighs 6 and values 10. B+C weighs 7 and values 11. A+D also weighs 7 and values 11. C+D has value 15 but weighs 9 and is illegal. Choosing only the single highest-value item D gives 8, which does not prove an optimal subset.
There are sixteen subsets including empty. Enumerate them or systematically rule out overweight combinations. The maximum here is 11, with two distinct selections A+D and B+C. Accept both; filling capacity is not the goal unless it also gives the highest value. Finish with one optimal selection and a reason no legal alternative beats it.
A class of thirty can use ten trios with selector, weight checker and value checker. Numbers are abstract marks rather than a prescription for children to carry loads. The technical reference defines the 0–1 choice, not evidence about this age group. For another capacity, create a new answer rather than reusing 11.