### Delivery Challenge

Dan and Dave have just finished sealing up three gigantic crates full of playing cards. One crate is full of blue decks, one crate is full of red decks and one crate is a mix of half red and half blue. Suddenly, they realize that all the crates are mis-labeled. The tricky thing is, every pack of cards is wrapped and sealed in the same brown paper so it’s impossible to know what color the cards are without unwrapping the individual packaging. What is the fewest number of crates and packs of cards that need to be unwrapped for Dan and Dave to know which label belongs on which crate?

#### Congratulations to everyone who correctly solved this puzzle. We've posted all the answers below. Special shout-out to the first three of you who solved it and won a a hand-turned spinning top: Alain Lacourse, Arvind, and Siddharth Rode.

Solution: It is only necessary to open one deck from the crate labeled "mix of half red and half blue." Because every crate is mis-labeled, whatever color deck is opened from the mixed crate must be that color. Now that this is known, let's say it's the red crate, you will know that the blue labeled crate is actually the mixed crate and that the red labeled crate is actually the blue crate.

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## 45 comments

Write a commentDiego C.

431! :)

David Wiliams

2 crates and 3 packs of cards.

Henry Chopp

Take 1 deck from the Mixed crate only should be enough to solve it

Kyle Byrne

Only one crate and one pack, choose the crate labelled mixed , that labels wrong so whatever colour you find when you open they must all be that colour. The other labels can then be figured out from this e.g if you find the mixed label box is actually all red then the one labelled red must be blue and the one labelled blue must be mixed.

Matt

You would need to open 1 crate and 1 pack. Specifically if ALL the crates are mislabeled there are only two way they could ALL be mislabeled.

Assume R = Red crate, B = Blue crate, and M = Mixed.

If the Initial order is

R B M

then the only way all can be mislabeled is by shifting left or right one space

B M R

M R B

As long as the open the one labeled as Mixed they know they will get a solid color crate. So they only need to open one pack from that crate. If it is Blue they know the one labeled Blue is actually Red and the one labeled Red is actually Mixed. Like wise if the pack they open is Red, then they know the one labeled Red is Blue and the one labeled Blue is actually Mixed.

Greg

If all of the crates are mislabeled, there are only 2 possible scenarios. You should only have to open one crate and one pack.

Matthew Ring

Because all 3 crates are mislabeled, you know that the “red” crate could either contain blue cards or a mix, the “blue” crate could either contain a mix or red cards, and the “mixed” crate could either contain red or blue. So open one of the mixed crate, if it is red, you know that the blue crate must be the mixed crate, and therefore the red crate is the blue crate. If the mixed crate is full of blue cards, then you know the red crate is the mixed crate and the blue crate is full of red ones. So how many cards do you need to open? One.

Ahmad Rami

Just the one that’s labled red and blue and one deck from it, you already know it’s mislabeled so if it has blue for example then the one that says blue has red and the red has red and blue because they’re all mislabeled.

Greg

The fewest possible would be 2 crates and 3 packs, but that is best case scenario.

Kehsav Mongia

The minimum number crates that need to be opened are 2. You only need to unwrap 3 decks.

If the first crate you open is the mixed crate, then you can open 2 decks and if they happen to be different then you would know which the mixed crate is.

Then you only have to open one more crate and one more deck. Whatever colour deck is in that crate is what that crate contains. The last crate therefore must contain the other colour decks. If the deck in that crate is blue then it is the blue crate and the last crate is the red crate.

This is the best case scenario.

Georg Ye

Best case scenario:

Since you dont know in which decks are in which crate you just have to randomly pick one.

Lets say you luckily pick the one with the half red and half blue ones. And you unwrap the first deck and its a blue one. Then you unwrap another one from the same crate and its red. So you know that that crate is the crate with the half red and half blue one.

Then you just need to unwrap another deck from one of those 2 crates and and you know which decks are in the other crate.

And it is a total of 3decks.

Worst case scenario:

You unwrap 1 deck from 1 crate and it is red. You unwrap another 1 because you might get lucky and get a blue one. But it is a red one.

So you unwrap 1 deck from each crate, so you got 1 blue and 4 red decks.

You know that the crate where the blue one belongs to is the crate with the blue decks.

And lets say you open another one from the crate you first unwrapped 2 decks. And it is another red deck.

And even if the 3 red decks belong to the crate with the half blue half red decks, if you get unlucky and always unwrap the red decks you have to unwrap half of the decks that are in the crate + 1 more.

So in the worst cast you have to unwrap half of the decks of 1 crate + 2.

Thales M. Meier

You’ll only need to open up one crate, but you are gonna have to unwrap at most 1/2 of the decks inside the given crate. After that, you’ll be able to correctly label all 3 crates.

Christian Au

2!

Sam McTernan

8

Scott Greene

Half plus one packages of cards From two crates.

Siddharth Rode

You only need to open one deck in one box. Since you KNOW that all of them are mislabeled – open the crate labeled mix of red/blue. Whatever the deck you open, this must be the color of all the decks in the box. For example, say you open Red. The labeled mix box contains only red now.

Then you go to the box labeled the color you didn’t find – BLUE in this example. This box must be mixed, since it can’t be Red (thats the mixed labeled one) or Blue (since this box is labeled blue, it can’t be blue).

The third remaining box must contain only Blue, since thats the only option left.

Arvind

You need to open only one crate and open only one deck of cards! It’s known that all 3 boxes are mislabeled so pick the crate labeled half red and half blue. Open that crate and open one deck. Now there are a two possibilities:

-If the deck you open is a blue deck, then you know that all the decks in that crate are blue. This is because it can’t be reds and blues (since it’s mislabeled) and it obviously can’t be all reds since the one you opened is a blue deck.

-Similar to the above, if the deck you open is a red deck, then you know that all the decks in that crate are red. Same reasoning as above.

Next, if the deck you opened was blue then you head to the crate labeled red. You know it can’t contain only red decks since it is mislabeled. And it can’t contain only blue decks since that is the crate you opened first. So this crate must contain a mix of reds and blues. And you then know what the final crate contains.

The other possibility is just the opposite which is if the deck you opened from the first crate was red.

Thales M. Meier

You’ll only need to open up one crate, but you are gonna need to unwrap at least 1/2 of the decks inside the given crate. After that, you’ll surely be able to correctly label each crate.

Kole Rogers

Two crates and three packs

Alain Lacourse

You only need to unwrap 1 deck from the crate labelled MIXED.

Since it must be a crate full red or blue, if you pick a red, then label this crate RED. Then the other crates are labelled RED and BLUE and must be in fact BLUE and MIXED. Since the box labelled BLUE cannot contain only BLUE decks, it is in fact the MIXED one, and the one labelled RED is in fact the one with BLUE decks.

Same logic if you pick a blue deck first.