Problem
People needed for three shared birthdays
A year has 365 days, so every birthday is one of 365 dates. The birthdays may fall however they like. Three people must share a date, unavoidably. Find the smallest crowd that guarantees it.
Reasoning · Grade 5-2 The Pigeonhole Principle
Your answer
Watch out Three times 365 feels right, but each date holds two people safely — the answer is one past that.
How to solve
Strategy Draw a Diagram — Picturing 365 labelled boxes, one per date, and dropping each person into the box of their birthday turns a vague question about crowds into a question about how full boxes can get. The answer is not just a formula: I must build the biggest possible gathering in which 3 people never share a date, count it, and then add one person. A tiny version with only 3 dates lets me build that failing arrangement by hand and see why the count comes out as 2 per date.
1STEP 1
Turn people and dates into boxes
People become balls and dates become boxes.
Turning a question about a crowd into a question about how full a box gets makes it something you can count instead of something you have to imagine.
4.OA.A.3Draw A Diagram2STEP 2
Describe exactly what failure looks like
Failure means at most two per date.
The unluckiest gathering is the one that spreads people out as evenly as possible, because bunching people up is what creates a triple.
4.OA.A.3Make A Systematic List3STEP 3
Test the idea on a 3-day year
With a three-day year the answer is 7.
3 × 2 + 1 = 7
With only 3 dates you can write the whole failing gathering out and see it fail, which shows the same recipe will work when there are 365 dates.
4.OA.A.3Solve An Easier Related Problem4STEP 4
Build the largest failing gathering for 365 days
For 365 days the largest failing crowd is 730.
365 × 2 = 730
365 twos is exactly what multiplication counts, and doubling 365 is easy: 300 doubles to 600, 65 doubles to 130, and 600 + 130 = 730.
4.NBT.B.5Make A Systematic ListThe largest gathering that can still fail has exactly two people on each of the 365 dates.
Why?
Filling every date to its limit is the most people that can be seated without three ever sharing a box.
🧱More items than boxes forces a shared boxn boxes can hold at most n items one apiece, so item number n plus one has nowhere new to go and must share.
Why?
Exhibiting one such gathering proves that any smaller number cannot be promised, so the bound is not merely a guess.
🧱One counterexample refutes an always-claimA claim about every case dies the moment you exhibit one case where it fails, and exhibiting is easier than arguing.
5STEP 5
Show 730 people is not enough
730 people cannot guarantee it.
730 people, every date carrying at most 2 → no triple
Showing one real gathering that fails is what proves the smaller number is not safe - a rule with no example behind it is where off-by-one mistakes hide.
4.OA.A.3Make A Systematic List6STEP 6
Add one more person
One more makes 731.
365 × 2 + 1 = 731
365 boxes holding 2 each can only hold 730 people in total, so person number 731 has to land in a box that already holds 2.
4.OA.A.3Draw A DiagramAnswer
731 people
365 × 2 + 1 = 731
The answer is a number of people, so it must be a whole number, and 731 is. It also has to be bigger than 365 x 2 = 730, since a crowd of only two-per-day already avoids a triple, and it should not be much bigger than that, since one extra person is all it takes. Both halves of the guarantee check out: 730 people can be arranged with no shared triple (2 on every date), and 731 people cannot. As a cross-check, the same recipe applied to 2 people sharing a birthday gives 365 x 1 + 1 = 366, which is the familiar answer, so the method is behaving correctly.
Takeaway
First build the unluckiest crowd you can - two people on every single date - then add one more person, because that person has to land on a date that already has two.
- Turn people and dates into boxes
- Describe exactly what failure looks like
- Test the idea on a 3-day year
- Build the largest failing gathering for 365 days
- Show 730 people is not enough
- Add one more person
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