Problem
Best and worst case draws for a pair
A drawer holds socks in 4 colours, 6 of each. They are pulled out one at a time without looking. Two socks of one colour are wanted. Find the fewest and the most draws needed.
Reasoning · Grade 5-2 The Pigeonhole Principle
Your answer
Watch out Six of each is a distraction: only the number of colours decides.
How to solve
Strategy Make a Systematic List — Best case and worst case are both questions about actual runs of draws, so the safe way to answer them is to write down a real run for each and check it. For the best case I list the luckiest run and see it works. For the worst case it is not enough to say "5 is the answer": I have to show a run of 4 draws that still leaves me with no pair (that is the witness that 4 is not enough), and then show that a 5th draw can never escape. Shrinking the problem to 2 colors first makes the shape of that argument obvious, and drawing one box per color makes it clear why 5 socks cannot all be different.
1STEP 1
Rule out 1 draw
One sock can never make a pair.
A pair means two things, so you cannot have one before you have taken two socks - this is counting, not luck.
2.OA.A.1Make A Systematic List2STEP 2
Show 2 draws can be enough (the luckiest run)
With luck 2 socks are enough.
gray, gray → pair after 2 draws
Writing one concrete lucky run down proves it is possible - no argument needed, you can point at it.
2.OA.A.1Make A Systematic List3STEP 3
Try the same question with only 2 colors first
With two colours the worst case is 3 socks.
gray, blue, ? → 2 + 1 = 3
A version small enough to write out completely shows exactly where the "plus one" comes from, and the same reason will work for 4 colors.
2.OA.A.1Solve An Easier Related Problem4STEP 4
Exhibit an unlucky run of 4 draws with no pair
With four colours 4 socks can still miss.
gray, blue, white, yellow → no pair yet
The only honest way to show a number is too small is to write down a run where it fails, and here the failing run is short enough to list in full.
4.OA.A.3Make A Systematic List5STEP 5
Show the 5th sock must always make a pair
The fifth sock always completes a pair.
1 + 1 + 1 + 1 = 4 ＜ 5
Four boxes with at most one sock each can only account for four socks, so a fifth sock has nowhere new to go - that is just counting the boxes.
4.OA.A.3Draw A DiagramThe fifth sock must always make a pair, because four boxes cannot hold five socks one apiece.
Why?
With four colours available, four socks can at most take one colour each, so a fifth sock has nowhere new to go.
🧱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?
Every way of avoiding a pair has been ruled out, so no unlucky run of five can exist however the draws fall.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
6STEP 6
Put the two answers together
So the answers are 2 and 5.
smallest = 2, largest = 4 + 1 = 5
Best case and worst case are two ends of the same list of possible runs, so answering both finishes the problem.
4.OA.A.3Make A Systematic ListAnswer
2, 5 socks
4 + 1 = 5
Both answers are counts of socks, so they must be whole numbers between 1 and 24, and 2 and 5 both are. The smaller answer must not exceed the larger one, and 2 is less than 5. The number 5 also passes the two tests a guarantee has to pass: there is a real 4-sock run with no pair (gray, blue, white, yellow), and there is no possible 5-sock run with no pair, because 4 colors can supply at most 4 all-different socks. Notice that the number 6 of socks per color never entered the arithmetic - it only had to be at least 2 so that a lucky pair exists and at least 1 of each so the unlucky run is possible.
Takeaway
To be sure of a match, first imagine the unluckiest run - one sock of every color - then take one more sock, because that one has nowhere new to go.
- Rule out 1 draw
- Show 2 draws can be enough (the luckiest run)
- Try the same question with only 2 colors first
- Exhibit an unlucky run of 4 draws with no pair
- Show the 5th sock must always make a pair
- Put the two answers together
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