Dalton's password

Logic Level 3

Dalton has a 6-digit combination lock. He gives it to his friends Albert, Bob, and Cheryl. He tells Albert the first two digits of the password, Bob the sum of all the digits, and Cheryl the last two digits. Then he gives them the following numbers and tells them that his password is one of them:

751846	741692	569843
563748	752746	814561
894362	813627	564932
721492	753846	748532

They then have the following conversation:

Albert: "I don't know the number. Do you, Bob?"
Bob: "Even I don't know the number. Cheryl?"
Cheryl: "No, I don't know it. Do you know now, Albert?"
Albert: "Not yet."
Bob: "I know it!"
Albert and Cheryl: "So do we!"

What is the password?

The answer is 753846.

1 solution

Stephen Mellor
Jun 16, 2018

We start off with the numbers shown below as possibilities:

$\color{#333333}{751846}$	$\color{#333333}{741692}$	$\color{#333333}{569843}$
$\color{#333333}{563748}$	$\color{#333333}{752746}$	$\color{#333333}{814561}$
$\color{#333333}{894362}$	$\color{#333333}{813627}$	$\color{#333333}{564932}$
$\color{#333333}{721492}$	$\color{#333333}{753846}$	$\color{#333333}{748532}$

Albert doesn't know the password means that the first two digits cannot be unique to a single password. This means that the password cannot be either of the ones labelled red below:

$\color{#333333}{751846}$	$\color{#333333}{741692}$	$\color{#333333}{569843}$
$\color{#333333}{563748}$	$\color{#333333}{752746}$	$\color{#333333}{814561}$
$\color{#D61F06}{894362}$	$\color{#333333}{813627}$	$\color{#333333}{564932}$
$\color{#D61F06}{721492}$	$\color{#333333}{753846}$	$\color{#333333}{748532}$

$\color{#333333}{751846}$	$\color{#333333}{741692}$	$\color{#333333}{569843}$
$\color{#333333}{563748}$	$\color{#333333}{752746}$	$\color{#333333}{814561}$
	$\color{#333333}{813627}$	$\color{#333333}{564932}$
	$\color{#333333}{753846}$	$\color{#333333}{748532}$

Bob doesn't know the password so the sum of the digits cannot be unique to a single password. Here are the sums of the remaining passwords with the unique ones highlighted in red. (Note that we are assuming that they all have deductive skills so Bob's information is based off what Albert has just said, and we don't need to consider the already removed passwords):

$\color{#333333}{31}$	$\color{#333333}{29}$	$\color{#D61F06}{35}$
$\color{#333333}{33}$	$\color{#333333}{31}$	$\color{#D61F06}{25}$
	$\color{#D61F06}{27}$	$\color{#333333}{29}$
	$\color{#333333}{33}$	$\color{#333333}{29}$

$\color{#333333}{751846}$	$\color{#333333}{741692}$
$\color{#333333}{563748}$	$\color{#333333}{752746}$
		$\color{#333333}{564932}$
	$\color{#333333}{753846}$	$\color{#333333}{748532}$

Cheryl doesn't know the password means that the last two digits cannot be unique to a single password. This means that the password cannot be either of the ones labelled red below:

$\color{#333333}{751846}$	$\color{#D61F06}{741692}$
$\color{#D61F06}{563748}$	$\color{#333333}{752746}$
		$\color{#333333}{564932}$
	$\color{#333333}{753846}$	$\color{#333333}{748532}$

$\color{#333333}{751846}$
	$\color{#333333}{752746}$
		$\color{#333333}{564932}$
	$\color{#333333}{753846}$	$\color{#333333}{748532}$

Albert still doesn't know the password means that the first two digits cannot be unique to a single password. This means that the password cannot be either of the ones labelled red below:

$\color{#333333}{751846}$
	$\color{#333333}{752746}$
		$\color{#D61F06}{564932}$
	$\color{#333333}{753846}$	$\color{#D61F06}{748532}$

$\color{#333333}{751846}$
	$\color{#333333}{752746}$
		$\color{#FFFFFF}{564932}$
	$\color{#333333}{753846}$	$\color{#FFFFFF}{748532}$

Bob now knows the password means that the password must have a unique digit sum. Here are the digit sums again:

$\color{#333333}{751846 = 31}$
	$\color{#333333}{752746 = 31}$
		$\color{#FFFFFF}{564932}$
	$\color{#333333}{753846 = 33}$	$\color{#FFFFFF}{748532}$

This means that Dalton's password must be $\boxed{753846}$ . Albert and Cheryl wouldn't be able to distinctly tell between the last 3 passwords by themselves, but they know that Bob knows it, allowing them to work it out as well.

Dalton's password

The answer is 753846.

1 solution

0 pending reports