Optimization — Greatest & Least
Name the objective, name the constraint — the best answer lives at the edge where they meet.
1 Explore — try these first
Try before you watch. Pick a level below and give the problem an honest try on paper first — wrong turns and all. Then open the video to see the trick. Every level rides one habit: Name the OBJECTIVE you’re pushing and the CONSTRAINT that stops it — the best answer sits at the EDGE where they meet.
★ · Objective vs Constraint
A relief team has 17 blankets and 14 water bottles. Every complete comfort kit needs 3 blankets and 2 water bottles. They also pack meals into crates that hold 5, 10, or 20 meals each, and they must deliver exactly 95 meals using the fewest crates. For each scenario, decide whether the question is asking for the GREATEST or the LEAST, name the objective (the thing being pushed), and name the constraint (the limit that stops it).
L0 · Bottleneck Rule
A relief packing line builds identical comfort kits. Each kit needs exactly 2 blankets and 3 water bottles. The warehouse has 17 blankets and 14 water bottles. What is the greatest number of complete kits the team can pack?
L1 · Take the Biggest First
Relief meals are shipped in crates that hold 5, 10, or 20 meals. To deliver exactly 95 meals, what is the least number of crates the team can use?
L2 · Squeeze the Rest
Seven relief grants are given out. Each is a different positive whole number of dollars, and together they total $350. One grant is exactly $100. What is the greatest amount a single grant could be?
L3 · Round Up to Guarantee
A field report says about 30 people will arrive (rounded to the nearest 10). Each person needs 2 meals. What is the smallest number of meals to prepare to guarantee no one falls short?
L4 · Shape the Extreme
A rectangular garden has whole-number side lengths in metres and a perimeter of exactly 24 metres. The length must be greater than the width. What is the greatest possible area of the garden?
L5 · Test the Boundary
A relief truck carries at most 100 kilograms. A crate of food weighs 6 kg and helps 15 families; a crate of water weighs 8 kg and helps 16 families. The team must pack at least 4 crates of food and strictly more water crates than food crates. What is the greatest total number of families helped?
L6 · Corner of the Region
Two kinds of relief kits are built. Each Type A kit uses 4 packs of supply X and 3 packs of supply Y and helps 4 people; each Type B kit uses 3 packs of X and 5 packs of Y and helps 5 people. The team has 27 packs of X and 34 packs of Y, and must build at least 2 of each kit. What is the greatest number of people that can be helped?
L7 · Optimization Detective
A relief station has 25 identical care packs to hand out. Every family that receives anything must receive a different odd number of packs. What is the greatest number of families that can each receive packs?
2 Learn — watch the solutions
Gave each one a real try? Now watch the trick. (Stuck is fine — that's the point.)
★ · Greatest or Least? Name the Edge
Peek the trick — Circle the superlative
Whenever a problem uses greatest, least, most, smallest, at most, or at least, name the one quantity to push (objective) and the limit that stops it (constraint); the answer is the last legal step before the limit.
L0 · The Scarcest Supply Runs Out First
Peek the trick — Min-of-floors, never a sum
When identical bundles each need fixed amounts of two or more supplies, the greatest number of COMPLETE bundles equals the smallest of floor(supply ÷ need-per-bundle) across every supply.
L1 · Take the Biggest First
Peek the trick — Greedy across nested sizes
When you need the fewest whole boxes to make an exact total and the box sizes nest (each divides the next), pack as many of the biggest size as fit, then the next size down, until the smallest closes the gap.
L2 · Shrink the Rest to Grow One
Peek the trick — Push every other part to its minimum
When a fixed total is split among all-different parts and you want one part as large as possible, make every other part as small as the rules allow; whatever is left over is your maximum.
L3 · Round Toward Safety, Never Short
Peek the trick — Un-round up, then ceil at every stage
When you must guarantee enough under rounded counts, push every input to its worst case (a nearest-10 report could be up to +4 higher), then take a ceiling at every indivisible step so safety always rounds away from zero.
L4 · Shape the Extreme
Peek the trick — Squeeze toward a square, or stretch into a line
When a fixed amount of border or tiles can be arranged many ways, the extreme value sits at the extreme shape: greatest area at a fixed perimeter is the rectangle closest to a square; greatest perimeter from connected tiles is the longest row.
L5 · Walk the Legal Edge
Peek the trick — List the few legal cases, push each to its limit
When a maximum has several integer constraints — a capacity cap, an at-least floor, and a comparison rule — list only the handful of legal integer cases, push each free variable to its limit, and the winner is the corner row.
L6 · Push the Right Corner
Peek the trick — Rewrite the objective to expose the lever
When you maximize a weighted total under two resource limits plus a floor, rewrite the objective so one variable becomes the visible lever, push it to the tighter of its two bounds, and the optimum lands exactly at that corner.
L7 · Pick the Right Lever (Detective)
Peek the trick — Diagnose first, then chain the right move
When no single rule closes the problem, diagnose which lever fits — min-of-floors (Bottleneck), round-up (Guarantee), boundary search, or a hidden structural cap like parity — and chain whichever moves the situation actually demands.
3 Master — practice on your own
Print the practice sheet and solve without the videos. Check your answers at the back — if one is wrong, the answer key names the trick so you know exactly which video to rewatch.
Download fresh practice problems PDF
