[dark_magician_92]

try solving this and give the reasoning as to how u solved it. i tried but cant get any insight to this problem except hit and trial

"This," started the guide, "is the Buttons Room."

The Druggar of Bongo Ghango - the chief of a large country by the River Ghango - looked around. "It's a nice room, but where are the buttons? The only ones I see are, ahem, the ones on your shirt."

"Your Highness. The buttons, which could start a nuclear armageddon in a matter of seconds, are there, behind that panel," replied the guide, while pointing at a large panel at the end of the room.

"How can that be??? Why? Anybody - a madman, for example - could come here and press those terrible buttons?"

"Your Highness, it is very safe actually: for every button there is a slot, where a magnetic card must be inserted to activate the corresponding button. No card, no button. To launch the missiles, all buttons must be activated and pressed, and only a handful of people have the magnetic cards, and each card contains a different code from all the other ones."

"But it's the same thing! Any of these persons could go completely ballistic and start a nuclear war."

"In that case, the only dangerous man is the President of the United States, as he is the only person that holds all the codes, which would allow him to press all the buttons. The other people that hold some codes are the Vice-President of the United States, the President of the Senate, the Secretary of State, the Chief of the Armed Forces, and the Dean of Harvard University. Each of these gentlemen holds an incomplete set of magnetic cards, and the distribution of the codes is such that, if the President of the United States is not available, the entire set of buttons can be activated by the Vice-President, together with anyone of the other four men. If both the President and the Vice-President of the United States are unavailable, the buttons can still be activated by any three of the other four men. Therefore, to launch the missiles, it is needed either the President, or the Vice-President plus anyone of the other four men, or any three of the other four men."

"What if someone tried to press randomly many buttons, one after the other?" asked the Chief.

"Nothing would happen with the missiles, but the room would fill up with a narcotic gas, and an alarm would alert the guards and the CIA."

"So, how many buttons are there, and how are they distributed between the Vice-President and the other four personalities?"

And that's the question we'll ask the reader: what is the minimum number of buttons, and how are they distributed?

[Guest]

I get 7 total cards distributed as follows

VP - cards 2 through 7

SOS - cards 1 through 4

CoAF - cards 1,2,5,and 6

PoS - cards 1,3,5, and 7

Dean - cards 1,4,6, and 7

Reasoning -

1) all of the four others must have the card that the VP is missing since he can combine with any of them to launch the missles.

2) all other cards must be held by a minimum two people (three of four together leaves one out - one of them still needs it).

So I did all combos of 2 people holding a card (A-B, A-C, A-D, B-C, B-D, C-D) to come up with six combos for the second one. Add in the card they all carry and I get 7 total

[Molly Mae]

try solving this and give the reasoning as to how u solved it. i tried but cant get any insight to this problem except hit and trial

"This," started the guide, "is the Buttons Room."

The Druggar of Bongo Ghango - the chief of a large country by the River Ghango - looked around. "It's a nice room, but where are the buttons? The only ones I see are, ahem, the ones on your shirt."

"Your Highness. The buttons, which could start a nuclear armageddon in a matter of seconds, are there, behind that panel," replied the guide, while pointing at a large panel at the end of the room.

"How can that be??? Why? Anybody - a madman, for example - could come here and press those terrible buttons?"

"Your Highness, it is very safe actually: for every button there is a slot, where a magnetic card must be inserted to activate the corresponding button. No card, no button. To launch the missiles, all buttons must be activated and pressed, and only a handful of people have the magnetic cards, and each card contains a different code from all the other ones."

"But it's the same thing! Any of these persons could go completely ballistic and start a nuclear war."

"In that case, the only dangerous man is the President of the United States, as he is the only person that holds all the codes, which would allow him to press all the buttons. The other people that hold some codes are the Vice-President of the United States, the President of the Senate, the Secretary of State, the Chief of the Armed Forces, and the Dean of Harvard University. Each of these gentlemen holds an incomplete set of magnetic cards, and the distribution of the codes is such that, if the President of the United States is not available, the entire set of buttons can be activated by the Vice-President, together with anyone of the other four men. If both the President and the Vice-President of the United States are unavailable, the buttons can still be activated by any three of the other four men. Therefore, to launch the missiles, it is needed either the President, or the Vice-President plus anyone of the other four men, or any three of the other four men."

"What if someone tried to press randomly many buttons, one after the other?" asked the Chief.

"Nothing would happen with the missiles, but the room would fill up with a narcotic gas, and an alarm would alert the guards and the CIA."

"So, how many buttons are there, and how are they distributed between the Vice-President and the other four personalities?"

And that's the question we'll ask the reader: what is the minimum number of buttons, and how are they distributed?

5 buttons, say A, B, C, D, and E

The Pres has all 5. ABCDE

VP has 4, ABC_E

the President of the Senate DAB

the Secretary of State, DBC

the Chief of the Armed Forces, DCE

the Dean of Harvard University - DAE

EDIT: Any three of the last 4 need to be able to access all 4, so the Dean is not unnecessary. And that throws out my whole plan to steal the cards...

Just pushed it up to 5.

Edited October 5, 2011 by Molly Mae

[dark_magician_92]

5 buttons, say A, B, C, D, and E

The Pres has all 5. ABCDE

VP has 4, ABC_E

the President of the Senate DAB

the Secretary of State, DBC

the Chief of the Armed Forces, DCE

the Dean of Harvard University - DAE

EDIT: Any three of the last 4 need to be able to access all 4, so the Dean is not unnecessary. And that throws out my whole plan to steal the cards...

Just pushed it up to 5.

nope, sorry, u missed out the fact that on not involving vp and President, 3 outta 4 are needed, in your solution there are few combos like PS and CAF are enough to have all keys.

EDIT: TYPO

Edited October 5, 2011 by dark_magician_92

[Guest]

MM, do you have?

two combos of the four men that can do it with only two of them? PoS and CoAF together and SoS and DoHU together?

Edit: nvm, dark magician answered my question

Edited October 5, 2011 by smoth333

[dark_magician_92]

yes U R ABSOLUTELY CORRECT

i had the same reasoning i.e. the two points but i dont understand why i didnt try to make the 6 combos, i was constantly thinking how to get the minimum no. anyhow thanks.

[Guest]

Sorry Molly, SoS and Dean can combine, as can other two...needs to have it be 3/4 needed

Edit - overkill

Edited October 5, 2011 by maurice

[Molly Mae]

Ah, I got you. Any three of the four other men can do it, but any two should not be able to do it. =/

[Guest]

Working backwards, we have:

For four men to each have cards so in combination they give a full code with any three: solution to this sub-problem is 4.

PS has cards AD

SS has cards AB

CAF has cards BC

DHU has cards CD

Now it is simple to add in the VP and Pres for, as previously pointed out by Moly Mae, The VP is only missing 1 card and all four of the others must have this card. So total is 5 cards.

President has ABCDE

VP has ABCD

PS has cards ADE

SS has cards ABE

CAF has cards BCE

DHU has cards CDE

Thinking of the 4 men only, each card must be held by 2 different men. There is no need for the card to be held by more than 2 and the sets held by each man must be unique.

[CaptainEd]

thoughtfulfellow, your solution permits PS and CAF to punch all the buttons, violating the requirement that it takes 3 out of the 4.

I think smoth333's solution is still the best.

Edited October 6, 2011 by CaptainEd

[Guest]

thoughtfulfellow, your solution permits PS and CAF to punch all the buttons, violating the requirement that it takes 3 out of the 4.

I think smoth333's solution is still the best.

I new it seemed too easy - I'll look at it some more

[Guest]

Working backwards, we have:

For four men to each have cards so in combination they give a full code with any three: solution to this sub-problem is 4.

PS has cards AD

SS has cards AB

CAF has cards BC

DHU has cards CD

Now it is simple to add in the VP and Pres for, as previously pointed out by Moly Mae, The VP is only missing 1 card and all four of the others must have this card. So total is 5 cards.

President has ABCDE

VP has ABCD

PS has cards ADE

SS has cards ABE

CAF has cards BCE

DHU has cards CDE

Thinking of the 4 men only, each card must be held by 2 different men. There is no need for the card to be held by more than 2 and the sets held by each man must be unique.

thoughtfulfellow, your solution permits PS and CAF to punch all the buttons, violating the requirement that it takes 3 out of the 4.

I think smoth333's solution is still the best.

Yes, after examining it further, I must concur that smoth333's solution is correct and has minimum possible cards. For someone that needs more explanation, this does adhere to my spoiler fo Further comments on though process, just needed more to avoid ability of two conspirators.