### 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?

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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 commentPranav Srivathsa

We open minimum two crates , As we open first crate then we have to open more than one of half of crate to make sure what color of cards in that crate . ( As here we make sure this crate have all red cards )

And

Same process for second crate to make what color of cards in that crate . ( As here we make sure this crate have all blue cards )

So, We can say that third crate have half of red and half blue cards without even opening the crate .

Therfore,

Minimum number of crates to be open = 2

And

Minimum number of packs of cards open = More than one of half of first crate + More than one of half of second crate

Finally , thank you so much for hosting this competition DAN AND DAVE!

brian

3 for the crates and 5 for the decks

Julia

Are the decks laid in any specific order in the crates?

Kade Havens

4 crates.

Mani Benavides

2 crates. 2 decks from each crate

Zak

two crates need to be opened.

Open the “mixed” crate. if the unwrapped deck is blue, then that is the blue crate (because its mislabeled). This crate would be the red crate if the unwrapped deck was red.

Open another crate and unwrap one deck. IF the crate opened is labeled “blue”,and the deck that is opened is blue, this is the mixed crate (as we have already deduced the blue crate. If it is red, then it is the red crate, as the last mislabeled crate is “red” and due to its mistake, can’t be the red crate. Thus, with two decks, you’ve figured out which crates are which and can reseal and send them out.

Ethan Choo

Since all the crates were labeled incorrectly, the crate labeled as ‘half blue and half red’ would have decks that are either all red or all blue. What crate the label belongs to can be determined by just unwrapping 1 deck in that crate.

Hence, Dan and Dave must check 1 deck in the crate labeled ‘Red and Blue’ (1 crate).

:P

Shoval

Only 1

You need to open the box who says mix and open one deck.

If the deck is blue it means that this is the blue decks and the one who says blue its red and the red is mixed because it says its mis-labeled .

If the deck is red it means that this is the red decks and the one who says its blue its the mix and the one who said red is blue.

Its a good one

Toby Hudson

My guess is one crate, one deck. Open the crate labelled HALF RED HALF BLUE and open one deck from it. If the deck is blue, this is the crate of blue decks and if the deck if red this is the crate of red decks. However, if the deck is blue, it means the crate labelled BLUE must contain red cards as EVERY crate is mislabelled. So therefore the crate labelled BLUE must contain red cards otherwise the crate labelled RED would contain red cards. Likewise if the deck opened in the HALF RED AND HALF BLUE crate is red, the deck labelled RED must contain blue cards and the box labelled BLUE must contain the mix of decks.

Piotr

At least one if they remember the sequence of packing they would know what is in other packages by opening one. They would open two to be sure what is in the third one. And no single package would be open if they really remembered the sequence of packing.

Brianyoo_

Zero. These cards are mystery decks sold during Christmas.

Isaac

5

Thomas

I think the answer is one crate and one pack of cards. Open up the crate labeled a mix of blue and red. Open up the first pack in the crate and since it can’t be a mixture – since all the crate labels are incorrect – it must have only that color in it. Eg. Blue.

Then the crate labelled with the color not pulled from the first crate’s pack of cards must have the mixed packs. It cannot have the color on the label because it’s mislabelled and of course it cannot have the color that the mixed crate has.

The last crate must have the remaining color packs.

Hopefully my logic works.

Luis Amaro

“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?”

1 Crate and 1 pack, since all the crates are mis-labeled they just have to grab a deck and unwrap it from the “both” crate, if its a blue deck, since it’s already wrong the label and it has a blue deck it can’t be the crate of both or red, then the crate with the “red” label should be one with the blue decks since the one “both” it’s already used and it’s already mislabeled with “red”, and the last one would be the red decks since its the only one left,

if they unwrap a red deck from the “both” label crate, since it’s already wrong the label and it has a red deck it can’t be the crate of both or blue, then the crate with the “blue” label should be one with the red decks since the one “both” it’s already used and it’s already mislabeled with “blue”, and the last one would be the blue decks since its the only one left

Luis Amaro

“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?”

1 Crate and 1 pack, since all the crates are mis-labeled they just have to grab a deck and unwrap it from the “both” crate, if its a blue deck, since it’s already wrong the label and it has a blue deck it can’t be the crate of both or red, then the crate with the “red” label should be one with the blue decks since the one “both” it’s already used and it’s already mislabeled with “red”, and the last one would be the red decks since its the only one left,

if they unwrap a red deck from the “both” label crate, since it’s already wrong the label and it has a red deck it can’t be the crate of both or blue, then the crate with the “blue” label should be one with the red decks since the one “both” it’s already used and it’s already mislabeled with “blue”, and the last one would be the blue decks since its the only one left

Gianluca

1 deck and 1 crate.

If the crates are mis labelled you open the one labelled mixed decks and one deck from that. Whatever colour is inside is the colour of the entire crate. And the other two crates must be switched for them to be mis labelled.

Dominic Quirante

1 crate and 1 box.

Brian

The fewest number of crates and decks that need to be unwrapped is 2 of each

Because there are 3 crates all with the same coloured decks inside, opening one deck from one crate and opening a deck from another crate will determine what those 2 crates hold inside them

Whichever coloured deck wasn’t unwrapped in the other 2 is automatically the other colour

Justin Cunningham

2 crates 3 decks

Greg

To clarify, if ALL of the boxes are mislabeled, all you need to do is open the crate labeled “Mixed” and open one pack of cards from that crate. If the deck is red, you know that the one labeled “Red” is actually blue and the one labeled “Blue” is actually mixed.

Michael Brandie

You would need to open 1 crate and from that crate you would need to open just over half of the packets (ie if each crate hate 40 decks you would open 21, so you would open 1 more than half). As all the crates are wrong then if you discover the crate you opened was 21 red (in this example meaning all the decks are red) and the label on this crate was mixed, then the crate marked red will be the blue cards leaving the crate marked blue with the mixed cards

Waverly lau

The minimum crates is two and in both you’d have to unwrap one more than half the pack of cards in the crate.

Rodolfo Eguiarte

Open the half red and half blue crate, since it’s misspelled you will end up with either red or blue cards. For now suppose you get red cards in that crate, now you are left with the crates “blue” and “red”. Open the crate labeled “red”, because it is mislabelled it will have blue cards and the “blue” crate will have the half red and half blue cards

Collin McLeroy

2 crates, and 2carss would have to be unwrapped.

Diego C.

431! By opening up to all decks except one, we are certain of the color of the package remained closed by counting how many decks of cards were revealed for each color.

I posted a little while ago the solution, here is the detailed answer! :)