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Stuck on this S1 question

tod

Member
In Jane's house one room has 7 lights in a row. Each light can either be switched ON or OFF. Jane makes the following two statements:
1. More than 2 Lights are OFF
2. Any OFF Light must not have exactly one adjacent ON light.

Given that 1 is False and 2 is TRUE, how many different unique arrangements are possible?

1. 5
2. 9
3. 12
4. 16
 

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A1

Admissions Speculator
Moderator
Statement 1 false means we can have 0 or 1 or 2 lights off = 7 or 6 or 5 lights on. I worked out 9 patterns (o on x off)

7 on = ooooooo
6 on combos noting no 1s allowed
0-6 = xoooooo
2-4 = ooxoooo
3-3 = oooxooo
5 on combos
0-0-5 = xxooooo
0-2-3 = xooxooo
0-3-2 = xoooxoo
0-5-0 = xooooox
2-0-3 = ooxxooo

This assumes mirror images (eg. 4-2 vs 2-4, 3-0-2 vs 2-0-3) don't count. If included they add up to 15 but is not in the answers.
 

sammy04

Member
its 16
because the combinations go upto more than 12 lol.

but i wouldnt waste ime on that question if its just 1 pointer, :)
 

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LMG!

Moderator
Staff Member of the Year 2019
Statement 1 false means we can have 0 or 1 or 2 lights off = 7 or 6 or 5 lights on. I worked out 9 patterns (o on x off)

7 on = ooooooo
6 on combos noting no 1s allowed
0-6 = xoooooo
2-4 = ooxoooo
3-3 = oooxooo
5 on combos
0-0-5 = xxooooo
0-2-3 = xooxooo
0-3-2 = xoooxoo
0-5-0 = xooooox
2-0-3 = ooxxooo

This assumes mirror images (eg. 4-2 vs 2-4, 3-0-2 vs 2-0-3) don't count. If included they add up to 15 but is not in the answers.
I don't necessarily agree that statement one being FALSE means there are 0, 1, or 2 lights on. I'd interpret it as meaning more than two lights are on... how do you come to your conclusion, @A1? What am I interpreting incorrectly?
 

LMG!

Moderator
Staff Member of the Year 2019
its 16
because the combinations go upto more than 12 lol.

but i wouldnt waste ime on that question if its just 1 pointer, :)
All questions are worth more than one point. Just to clarify, do you mean if there weren't multiple questions based on the same stimulus?
 

A1

Admissions Speculator
Moderator
I don't necessarily agree that statement one being FALSE means there are 0, 1, or 2 lights on(??). I'd interpret it as meaning more than two lights are on...
More than 2 lights on can make but does not guarantee the statement false. Like 4 on + 3 off, the latter makes it true.
 

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tod

Member
Perhaps a good place to start would be to provide your method and how you got the answer 5.
So this is the way I saw it.

From Rule 1, we say that 0,1 or 2 lights are off.
'Any OFF Light must not have exactly one adjacent ON light.'
If we take this as xxxxxxx, we can already exclude the off lights from being on either end of the sequence as this would mean there is only 1 adjacent ON light if the on light was 2nd in the sequence.

So you would be left with five possible places for the off lights.
xxxxxxx
x0xxxxx
x0x0xxx
x0xx0xx
x0xxx0x
xx0x0xx
xx0xx0x
xxx0x0x
xx0xxxx
xxx0xxx
xxxx0xx
xxxxx0x
....actually i just realised its a hec more than five :)

Guess the answer was 12 C
 
Last edited:

A1

Admissions Speculator
Moderator
'Any OFF Light must not have exactly one adjacent ON light.'
If we take this as xxxxxxx, we can already exclude the off lights from being on either end of the sequence
Oops my interpretation above of the rule 2 was incorrect. Indeed it means an Off light must be between two On lights, and 12 is the answer.
 

Kat92

(Student BNurs) Hopeful for JMP
I am thinking it could be 16 as I end up with 15 doing it both ways.
 

Mana

there are no stupid questions, only people
Administrator
Here's my attempt at it.

So if you have two statements:

1. More than 2 Lights are OFF - False (essentially, you have the statement that there are 0, 1, or 2 lights off)
2. Any OFF Light must not have exactly one adjacent ON light

for 0 lights off, there is 1 combination (all on)
for 1 light off, there are exactly five combinations (the single off light being anywhere except either end)
for 2 lights off, as per Rule 2, the off lights cannot be on either end and the off lights cannot be adjacent to each other. Basically this means in the middle 5 lights, you have (where 1 is an OFF light)
lights 4 apart x 1 10001
lights 3 apart x 2 10010 and 01001
lights 2 apart x 3 10100 01010 00101

This gives the requisite 12.
 

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