Select non-overlapping activities
Maximize the number of compatible equal-value intervals and audit the boundary rule.
Adult planning suggestion: about 9–12 · Logic & rule checking · Shape & spatial thinking
- Players
- Pairs
- Time
- 12–18 minutes
- Materials and quantities
- Eight interval cards, one 0–12 timeline and a selection table per pair.
Thirty pupils: fifteen pairs with selector/checker roles swapped for the second trace.
Prepare
The fictional intervals are A 0–4, B 1–3, C 3–5, D 4–7, E 5–8, F 8–10, G 7–11 and H 10–12. Times are abstract equal units, not pupils’ calendars. Every activity has equal value, no setup or travel time, and touching endpoints are allowed: a new interval may start exactly when the previous one ends. Write this boundary rule on the timeline.
Play
Select as many intervals as possible with no shared interior time. First try a set and check every neighbouring finish/start pair after arranging it in time order. Then test the earliest-finish method: consider intervals in increasing finish time and accept one when its start is at least the previous accepted finish. Skipping an incompatible interval does not move the current finish time. Keep excluded intervals visible for explanation.
Worked example
The checked order accepts B 1–3, C 3–5, E 5–8, F 8–10 and H 10–12, giving five. D is considered before E but its start 4 is less than the current finish 5, so D is skipped. F may follow E because 8 equals 8. Starting with the earliest-starting A is not a reason to assert an optimal answer; a greedy choice needs the correct criterion.
Finish and check
Finish after a valid five-card set and a trace of accepted/skipped intervals, or stop after fifteen minutes with the best audited set so far. The key verifies this finite instance by checking all 256 subsets; it does not ask pupils to enumerate them. Equal rewards matter: if activities gain different values, this earliest-finish rule does not automatically maximize total value.
Adaptation and practical limits
For thirty pupils use fifteen pairs. One chooses while the other checks boundaries; swap for the algorithm trace. Adult planning suggestion is roughly 9–12 with number-line support. The source is a university algorithms lecture, used only for the selection mechanism. Provide large timeline marks or lay cards on paper; there is no need to collect anyone’s real timetable or buy scheduling software.
Aim: Maximize the number of compatible equal-value intervals and audit the boundary rule.
Practice within this activity: Separate a start-time guess from a finish-time selection rule.
Adult observation: Ask why D is skipped while F is allowed at an equal endpoint.
Mechanism and adaptation: Our version uses unweighted interval selection by earliest finishing time. Examples and sheets are separately written; the earliest inventor is not established here.
Reference checked 25 October 2026: Princeton · Greedy Algorithms I — University lecture background for equal-value interval scheduling by earliest finish; no classroom outcome or weighted-task guarantee.
Goals describe observable activity choices, not measured long-term gains or institutional endorsement.
Free alternatives: Use paper or clean reused cardboard and a pencil. Copy the supplied diagrams and tables by hand; no specialist kit or purchase is needed.
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 interval cards
Equal value; no travel/setup; touching endpoints allowed.
Player timeline and selection trace
| Finish order | Accept / skip | Current finish |
|---|---|---|
| B | ____ | ____ |
| A | ____ | ____ |
| C | ____ | ____ |
| D | ____ | ____ |
| E | ____ | ____ |
| F | ____ | ____ |
| G | ____ | ____ |
| H | ____ | ____ |
Host / teacher checked example · separate
Thirty pupils: fifteen pairs with selector/checker roles swapped for the second trace.
Eight interval cards, one 0–12 timeline and a selection table per pair.
Keep this answer page separate. Read the full web rules and preview materials before leading; allow pauses, passing and supported writing.
Accept B,C,E,F,H: five intervals. Skip D because4<5; F follows E because8=8. All256 subsets checked; maximum5. Different values require a different optimization rule.
| Selected | Interval |
|---|---|
| B | 1–3 |
| C | 3–5 |
| E | 5–8 |
| F | 8–10 |
| H | 10–12 |
A declared one-unit cleanup variant · separate player exercise
| Plan | Finish+1 | Next start | Valid/count |
|---|---|---|---|
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
A declared one-unit cleanup variant · host check
With cleanup1 the maximum is3, checked over all256 subsets. B,D,F is valid:3+1≤4 and7+1≤8. B,E,H is also valid. B,C is no longer compatible because3+1>3. The original zero-cleanup maximum5 is unchanged.
Related activities
A declared one-unit cleanup variant
The original eight intervals still allow touching endpoints. On this separate sheet require a full extra unit after each chosen interval before the next begins. Do not silently apply this variant to the original five-card key. All activities still count equally.
Why an Earlier Start Can Fit Fewer Activities
Say which quantity you are maximizing
In Select non-overlapping activities, each fictional activity counts once. The aim is the greatest number of compatible cards, not the longest total occupied time or the earliest beginning. There is no travel, preparation gap or variable reward. A finish at time five can be followed by a start at time five. Put that endpoint rule beside the timeline before comparing suggestions. Otherwise two people may evaluate the same pair differently while each is following a reasonable but different interpretation of an ordinary real-world timetable.
Test an earliest-start suggestion honestly
The earliest-starting printed activity is A, running from zero to four. If you keep accepting compatible cards in start-time order, D from four to seven follows it, then G from seven to eleven. The remaining cards cannot add another compatible activity to that sequence. You have selected three. This schedule is valid; its problem is that another valid schedule contains more cards. Separating legality from quality keeps the comparison precise. Do not reject A simply because it differs from the answer key. Explain which later options it prevents and compare the resulting total.
Compare the finish-time decisions
In increasing finish-time order, first accept B from one to three. A finishes next but begins before B ends, so skip it without changing the current finish of three. Accept C from three to five; skip D because its start is four. Accept E from five to eight, then F from eight to ten. G starts before ten, so skip it. Finally accept H from ten to twelve. That gives five activities. Read the current finish after each decision so a rejected card does not accidentally move the boundary used for the next check.
Explain what an earlier finish preserves
Suppose a compatible plan begins with some activity X, while another available activity finishes no later than X. Replacing X with that earlier finisher does not push the start of the remaining plan later. That is the useful direction of comparison: finishing earlier preserves room afterward. Starting earlier alone offers no such protection, because an early activity may continue for a long time. The Hope College interval-scheduling explanation develops this exchange reasoning. Its own worked intervals are different; the five-card result above comes from the supplied PartyPageKit cards.
Do not substitute shortest duration without checking
A shorter activity can sit across a useful boundary. On a separate practice slip, write J from zero to three, K from three to six, and L from two to four. L is shortest, lasting two units. Yet choosing it blocks both J and K, leaving one selected activity. Choosing J followed by K gives two. This is a new three-card example, not an instruction to replace the eight printed cards. It shows why “shortest” and “earliest finishing” are different criteria even though both might sound like ways to leave spare time.
Keep a valid alternative available for comparison
Using the printed cards, A, D, F and H form a compatible four-activity plan. It contains a gap between seven and eight, which is allowed. A gap is not automatically an error, and filling more time is not the stated score. Compare this set with the five-card B, C, E, F, H plan by counting accepted cards, then checking every neighbouring finish and start. Retain both records so participants can explain the improvement. The smaller set can be a useful intermediate proposal rather than an answer that must be erased from the discussion.
Stop when the assumptions change
If a new version assigns different rewards, maximizing the card count may no longer maximize total value. Likewise, adding cleanup time changes compatibility at touching endpoints. Label such a version as a new problem and reconsider the rule instead of applying the old key automatically. For the printed equal-value task, finish with a checked selection and an explanation of accepted and skipped cards. A partner may place the cards while the other traces the boundary. This paper exercise illustrates one scheduling model; it does not inspect anyone's calendar or promise to optimize a real household's competing priorities.