Find the Objects, Drawers, and Guaranteed Count
Turn a crowded situation into a proof. Identify the objects, choose the drawers, build the greatest arrangement that can still fail, and add the one object that forces success.
The worked example calls them “apples” and “drawers”
The objects are the things being placed, chosen, or classified. The drawers are the categories they can enter. A drawer can be a score, birth month, card rank, color pair, or box occupancy.
Two guarantee formulas
Identify objects, drawers, and the overloaded drawer
Interactive drawer model
Move the sliders. The page distributes the objects as evenly as possible—the arrangement that makes the fullest drawer as small as it can be.
Balanced distribution
Checkpoint
Seventeen objects are placed into five drawers. At least how many objects must be in one drawer?
Need a hint? Start here
Try the most even distribution.
Show the reasoning
Five drawers can hold 3 each: 15 objects. With 17 objects, at least one drawer holds 4. The arrangement 4,4,3,3,3 shows you cannot force 5.
Work backward from the guarantee you want
Reverse guarantee builder
The “one more” proof
- Keep every drawer at no more than r−1.
- The greatest unsuccessful total is k(r−1).
- The next object has nowhere safe to go.
Checkpoint
How many objects guarantee at least four in one of six drawers?
Need a hint? Start here
How many fit if every drawer stays one below the target?
Show the reasoning
Six drawers can hold 3 each without reaching 4: 18 objects. One more gives the guarantee 19.
Worked example 5 — force a repeated card rank
Build the greatest safe draw
Take one card from every ordinary rank, then both different jokers. No ordinary rank repeats yet.
Why the next card forces success
After all 15 safe types have been used, every remaining card belongs to an ordinary rank already represented.
Checkpoint
Need a hint? Start here
Jokers do not count as an ordinary rank pair.
Show the reasoning
You can choose one of each of 13 ordinary ranks and both jokers: 15 cards with no ordinary pair. Card 16 must repeat an ordinary rank.
Worked example 6 — guarantee the multiset 2, 0, 0, 9
Inventory in the bag
Digit d appears on 10+d balls.
Find the largest way to fail
Draw balls without replacement. You may choose any four of the drawn balls and arrange them to read2009; their draw order does not matter. To avoid forming2009, at least one requirement must be missing.
| Failure condition | Most draws | Reason |
|---|---|---|
| At most one zero | 136 | All 135 nonzero balls + one zero |
| No digit 2 | 133 | All balls except the 12 twos |
| No digit 9 | 126 | All balls except the 19 nines |
Checkpoint
Need a hint? Start here
List every way the digits for 2009 could still be missing.
Show the reasoning
Failure can mean at most one zero, no 2, or no 9. From the total 145, those permit 136, 133, or 126 draws. The largest failure is 136, so 137 guarantees success.
Guided Practice 5 — strengthen a basic guarantee with the total
First use ordinary pigeonhole counting
The six possible occupancies—1 through 6 balls—are the drawers.
The brackets ⌈ ⌉ mean divide, then round upward if there is a remainder. For example, ⌈17 ÷ 5⌉ = 4.
This guarantees three boxes with the same occupancy, but the total of 64 gives more information.
Why three each cannot happen
If no occupancy appeared four times, all six occupancies would have to appear exactly three times:
There are actually 64 balls, so some occupancy appears at least four times.
Occupancy-frequency laboratory
Each input is the number of boxes containing 1, 2, …, 6 balls.
Checkpoint
Need a hint? Start here
The drawers are occupancies, not physical boxes.
Show the reasoning
There are six possible occupancies. If each occurred at most 3 times, 18 boxes require exactly 3 of each. That totals 63 balls. The 64th forces an occupancy to occur at least 4 times.
Transfer the principle to scores and birthdays
Test 23 transfer — integer scores
There are 10,000 test takers. Four fifths score at least 60. Scores are whole numbers and the maximum is 150.
Test 23 transfer — same birth year and month
Fifty students were all born in either 2014 or 2015. Their birth year–month falls into at most:
Checkpoint
Need a hint? Start here
Include both endpoints of a score range. Use the stated calendar years.
Show the reasoning
There are 150 − 60 + 1 = 91 integer scores; 91 × 87 = 7917 < 8000, so at least 88 match. Two calendar years give 24 year–month categories; 2 in each accounts for 48, so 50 force 3.
Combine lower and upper guarantees, then count pair types
Exactly how many girls?
Fifty-five contestants are divided into four nonempty groups. In every possible division, at least one group has more than two girls. This forces at least:
Any ten contestants include a boy, so ten girls cannot exist. Thus there are at most 9 girls.
Two-ball color-pair outcomes
With five colors, each person draws two balls and replaces both before the next person draws. There are at least two balls of each color. Each result is an unordered pair with repetition. Red–blue and blue–red are one type; red–red is allowed.
Sixteen people force two identical color-pair outcomes.
Checkpoint
Need a hint? Start here
Turn each statement about girls into a bound. For colors, ignore order.
Show the reasoning
The group condition forces at least 9 girls; any ten containing a boy permits at most 9 girls. There are 46 boys. Two-ball types are 5 matching colors plus 10 mixed pairs, so 16 people force a duplicate.
Use capacities in a general worst-case laboratory
Capacity-aware guarantee (draw without replacement)
For capacities c₁,c₂,…, the greatest draw avoiding r from one category is:
Two direct transfers
Colored balls: capacities 4, 7, 8; target 6.
Toys among six children: force four toys for one child.
Checkpoint
Need a hint? Start here
Each category can contribute at most target minus one, but never more than its supply.
Show the reasoning
For capacities 4,7,8 and target 6, the safe draw is 4+5+5=14: 15 forces six. To force four toys in one of six categories, 6×3+1=19 toys suffice.
Try a fresh problem
Seven shelves receive books. What is the least number of books that guarantees at least six on one shelf?
This fresh problem has its own checkpoint. Your written explanation is saved for comparison and is not automatically graded.
Worked explanation for the fresh problem
Seven shelves with five books each hold 35 without six together. Book 36 forces six.
Guaranteed-count workshop
Need a hint? Start here
Revisit the mission connected to each question.
Show the reasoning
Answers:4,19,16,137,4,88,3,16. For each minimum threshold, build a draw or distribution that falls one short:18objects,15cards,136balls,or15different pair types. The64-ball occupancy total strengthens3matching boxes to4.
Exit ticket and certificate
Objects & Drawers Guarantee Architect
This certificate recognizes
for identifying objects and drawers, building worst-case arrangements, and proving sharp guarantees.
Lesson 23.1 completed
Need a hint? Start here
Solve each item without using the answer shown in an earlier example.
Show the reasoning
28objects in9drawers force4. Seven drawers holding4each can hold28, so29forces5. Supplies4,7,8 allow at most4+5+5=14without6matching, so15forces it. Four colors give4matching-color types and6mixed types:10types, so11people force a repeated type.
Optional reflection
Saved, but not automatically graded.
Exercise notes and lesson scope
This page develops the chapter introduction’s “apples and drawers” language, original Examples 5 and 6, Guided Practice 5, and selected direct-guarantee exercises. It also uses four transfer problems from Test 23. The ceiling formula, capacity laboratory, unordered-pair matrix, tight examples, and automatic feedback are added learning supports.