T = Truth guy, L = Liar, B = Both truths and lies guy

6 Permutations

1 2 3
T L B
T B L
L T B
L B T
B T L
B L T

basically you wanna ask questions to either T or L because asking B is pointless. So try to ask questions that will make person is spot 1 2 or 3 either a T or an L

Q1: "First guy, is T is to the right of B or is L is in the middle?" (i.e. if one is true the answer is yes)

if YES, then it's

1 2 3
T L B
L T B
B L T
B T L

NOTE: in this case column 2 is either Ts or Ls so you're gonna ask him the next question

if NO, then it's

1 2 3
T B L
L B T
B L T
B T L

NOTE: in this case column 3 is either Ts or Ls so you're gonna ask him the next question
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So, if Q1 = yes, Q2 could be: "Second guy, are you the 'Both' guy"

if Yes, he's lying, so it's

1 2 3
T L B
B L T

if No, then he's telling the truth, so it's

1 2 3
L T B
B T L

Now if Q1 = no, Q2 could be: "Third guy, are you the 'Both' guy"

If Yes, he's lying, so it's

1 2 3
T B L
B T L

If No, he's telling the truth, so it's

1 2 3
L B T
B L T

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Q3 is the easiest, now that you know for sure that one of the spots is a liar or a truth guy so you just ask him who one of the other guys is and you'll know all 3.

for example, if you know guy lets say you ask Q1 and he says yes, and then ask Q2 and he says yes again. Therefore you know that guy 2 is a liar. Now ask him if guy 3 is a truth teller. If he says no, then you know guy 3 is the truth teller and guy 1 is the both guy.

you can do that for all the rest of them

because there aren't multiple answers that add up to the same house #

9 + 2 + 2 = 13 and 6 + 6 + 1 = 13

you could ask, which of the other 2 is more likely to tell a lie. If he's the liar, he will point at the truth teller. If he's the truth teller, he will point at the liar. If he's random, you're not going to be asking him the next Q anyways so you're safe.

To answer the first question:

Seems to me, that you can just ignore the misses, since it doesn't matter who you would be shooting at then.

So all it comes down to is kill man 2 -> get killed 100%, kill man 3 -> get killed <100%.

Will look at the exact %'s unless this is wrong.

man 1 should miss on purpose, I think

After miss by man 1:

Scenario 1, man 2 shoots man 3 (67%):

1/3+(2/3*1/3*1/3)+(2/3*1/3*2/3*1/3*1/3)+(2/3*1/3*2/3*1/3*2/3*1/3*1/3)....~43%

Scenario 2, man 2 misses is then killed by man 3 (33%):

Man 1 gets one shot to kill man 3=33%

Total chance of survival: (67%*43%)+(33%*33%)~39.8%

If he shoots at man 3 he has (67%*39.8%)+(33%*33%*43%)~31.3%

Man 1 wins 39.8%
Man 3 wins: (1/3*2/3) ~ 22.2%
Man 2 wins: (2/3*57%) ~ 38%

Yes man 1 should intentionally miss, and mute's analysis is right up to a sig fig or so. It is trickier to compute this if there is an arbiter who will shoot you if you miss intentionally. So there's my bonus add-on. If you try to compute this, you should get: