Keep the useful bits, human.
Beginner field notes
- Capacity is the most one bin may hold in the stated unit. In the lesson’s capacity-10 bin, a size-4 item leaves 6 units free.
- Each item stays whole and is placed exactly once. Splitting or silently dropping an item is not a valid way to make this model fit.
- A load of 10 meets a capacity of 10; a load of 11 exceeds it. Check every bin’s load, not just the total across all bins.
See a worked example
Check a placement before asking whether it is best
Sample only. Nothing here runs on your computer.
OFFLINE Lantern Archive paper placement; no boxes, machines or files are actually changed.
Four distinct, whole items: A size 4, B size 4, C size 6, D size 6. Each item must be assigned exactly once and cannot be split. Sizes are positive integer units of one additive resource; every bin has capacity 10. A bin is legal when its total is at most 10, so equality is allowed. Ignore shape, overhead, timing, CPU/memory combinations, affinity and every other constraint.
Start with no bins. Give each newly opened bin the next label P1,P2,P3. Once an item is assigned, do not move or remove it. There are enough identical bins; each item fits an empty bin.
First-fit rule: take items in the stated order. Check already opened bins in their opening order; put the item in the first whose remaining capacity is at least its size. If none fits, open a new bin and put it there.
Use input order A4, B4, C6, D6. After each item, show the selected bin, contents, load and remaining capacity. Does every final bin satisfy the limit?A4: open P1. P1=[A], load 4, remaining 6.
B4: P1 is the first fit. P1=[A,B], load 8, remaining 2.
C6: P1 remaining 2 is too small; open P2. P2=[C], load 6, remaining 4. P1 remains unchanged.
D6: P1 remaining 2 and P2 remaining 4 are too small; open P3. P3=[D], load 6, remaining 4.
Final bins: P1=[A,B] load 8/10; P2=[C] load 6/10; P3=[D] load 6/10. All four items occur exactly once; all loads are within 10. Three bins are used.
This is a valid placement under the given rule; the capacity check alone does not prove it uses the fewest bins.Track each bin independently and subtract its current load from 10 before assigning the next whole item. The three-bin result follows the supplied input order and first-fit rule. It satisfies capacity and exact-once placement; it does not establish a minimum, and no actual bin or resource is created.
Visual
The reviewed description and complete diagram source are available here.
More of this diagram is available inside its frame. Focus the frame and use the arrow keys to pan.
Read the complete diagram source
This is the source for the reviewed diagram, not a command to run.
flowchart LR
A["A size 4"] --> P1["P1: A+B; load 8 of 10"]
B["B size 4"] --> P1
C["C size 6"] --> P2["P2: C; load 6 of 10"]
D["D size 6"] --> P3["P3: D; load 6 of 10"]
P1 --> V["All items once; each bin within capacity"]
P2 --> V
P3 --> VSources and full explanation
Notes sources · Beginner
AI source review · 2026-10-17
Reviewed explanation
A capacity limit says how much one container can hold under a stated rule. A placement says which item goes into which container. In the fictional Lantern Archive, a paper bin holds at most 10 units of one resource. An item labeled A uses 4 units. Put A in an empty bin and its load becomes 4, leaving 6. No terminal, programming language or network lesson is needed to follow that arithmetic.
We must first define what fits. Items are whole and indivisible here, so a 6-unit item needs 6 free units in one bin. Two separate bins with 2 and 4 free units do not give that item a usable 6-unit slot. A total of 10 meets the limit exactly; a total of 11 does not. Every item must be placed exactly once, and all sizes use the same additive unit.
A constraint tells us which placements are allowed. An objective tells us which allowed placement we prefer. Keeping every bin within 10 is the capacity constraint. Trying to use the fewest bins is a possible objective, but passing a capacity check is not proof that the objective is best. A different legal arrangement can use a different number of bins.
To make the paper trace unambiguous, we choose first-fit: process the listed items in order and scan existing bins in their opening order. Use the first that has enough remaining capacity, or open the next labeled bin if none does. Earlier assignments stay fixed. A and B take 8 units together; the following 6-unit items each need a new bin under that rule.
These are mathematical bookkeeping rules, not a description of a live storage system or scheduler. Our one-dimensional model omits shape, overhead, timing and combinations such as CPU plus memory. Such extra constraints must be modeled separately if they matter. Reading the worked trace assigns no real resource, and the number of bins supplies no performance or cost guarantee.
- NIST Dictionary of Algorithms and Data Structures: bin packing problem
Defines constrained packing and the common fewest-fixed-capacity-bins objective, while distinguishing variants such as dimensions and resource measures. It does not equate this one-dimensional example with a real scheduler.
- Google OR-Tools: The Bin Packing Problem
The official model assigns each item to exactly one bin, limits its additive load to the common capacity and minimizes bins used. The lesson borrows these mathematical constraints, not solver code, runtime results or an operational deployment procedure.
- Mathematical Optimization book: Bin packing and cutting stock problems
The method authors describe non-increasing-size preprocessing followed by the first open bin that fits, opening another only when needed. They explicitly distinguish a heuristic from guaranteed optimization. The chapter also illustrates an instance-specific total-capacity lower bound. Only these explanations are used; its code/solver setup is not executed.
AI source comparison recorded 2026-10-17. Human learner validation is pending.
Quick quiz · one question
Follow capacity checks without moving items
Sample scenario. Offline Lantern Archive paper procedure, distinct from an automatic optimizer. One additive resource, capacity 10, whole items A size 4, B size 4, C size 6, D size 6. Every item fits alone. Open only the next numbered container when a listed valid attempt first uses it. Refuse an overfull attempt and retain its unplaced item; retry is allowed. No split, move, removal or automatic repacking.
- Paper capacity
Each container holds at most 10 integer units of one additive resource. Items are whole and each fits alone. No splitting, moving, removal, hidden repacking, real allocation or other constraint.- Distinct items
A: size 4; B: size 4; C: size 6; D: size 6- Trial manual
Only the listed manual targets apply. Open at most the next numbered container when needed; refuse overfull tries and retain the unplaced item for a later attempt.- manual: attempt 1
Try whole item A in P1.- manual: attempt 2
Try whole item B in P1.- manual: attempt 3
Try whole item C in P1.- manual: attempt 4
Try whole item C in P2.- manual: attempt 5
Try whole item D in P1.- manual: attempt 6
Try whole item D in P2.- manual: attempt 7
Try whole item D in P3.
Follow only the seven stated placement attempts. Which containers exist at the end, and what happens to an item after an overfull attempt is refused?
Why this answer?
Answers stay on this device.
Sources and exercise scope
- NIST Dictionary of Algorithms and Data Structures: bin packing problem
Defines constrained packing and the common fewest-fixed-capacity-bins objective, while distinguishing variants such as dimensions and resource measures. It does not equate this one-dimensional example with a real scheduler.
- Mathematical Optimization book: Bin packing and cutting stock problems
The method authors describe non-increasing-size preprocessing followed by the first open bin that fits, opening another only when needed. They explicitly distinguish a heuristic from guaranteed optimization. The chapter also illustrates an instance-specific total-capacity lower bound. Only these explanations are used; its code/solver setup is not executed.
Original synthetic exercise. AI comparison against the reviewed lesson recorded on 2026-10-17; the references above state their scope. Human learner validation has not been run. A result here records this attempt; it does not establish mastery.
Save or adjust this lesson
Make this lesson yours
Save it for later, or tell your robot what would help.
This correction stays on this browser. Browse all concepts if the topic is missing.
Familiarity is your own assessment. To use it in your coding tool, download your updated learning profile.
Choices are saved on this browser. They do not sync to your account or coding tool.