The poisoning in the hospital

Isn't it simpler like this? Both the daughter and the housekeeper said they saw the same 2 person, which means at least one of them should've seen each other, which they did not mention, so from here the liar is either daughter or housekeeper, the fact that there's only one liar means that the wife is telling the truth, thus the liar is the housekeeper.

Wife, daugheter, manservant and lawyer tell the truth. Because wife and daughter saw the other, manservant and lawyer saw the other. Wife and lawyer were not in the hospital at the same time because none of them saw the other. Moreover the daughter saw the wife and the lawyer so she must be in the hospital before the wife left and after the lawyer entered. The son and the housekeeper said that they saw the wife and the lawyer so they must be in the hospital at the same time the daughter was. But the housekeeper and the daughter didn't see the other, the housekeeper is liar and he is the culprit

To start solving the question we should take all the words and write it as simply as possible. The way I did it was substitute each name with a variable.

Step 1: Write out the information. Let W = wife D = daughter S = son M = manservant H = housekeeper L = lawyer So the information would be ( let : be the symbol for saw) W: d & s D: w & l S: d & m M: h & l H: w & l L: s & m

Step 2: Start solving the question. The easiest way to solve this problem is to realize that if one person saw another person and the other person also saw that person then that person is telling the truth. But we should also take into consideration the time of arrival (or overlap of time spent). The best way to do this is to take into consideration that all are saying the truth.

W: left before the manservant arrived D: left before manservant arrived S: left before housekeeper arrived L: left before housekeeper arrived M: left after housekeeper arrived H: left last

Step 3: Check the suspects statements. Keep in mind only the culprit lied.

Lets say the housekeeper lied, after all her statement has no possible way of being true since the wife left even before she came, and so did the lawyer. So we found the culprit.

On order to check we didn't mess up we can check the statements of the other suspects.

Lets say the daughter lied, then we need to take the opposite of her statement. "I did NOT see the wife." Which is not possible since the wife saw the daughter. So daughter is NOT the culprit, nor is the wife.

Lets say the lawyer is the culprit. The his statement becomes "I did not see the son or manservant" both of whom saw the lawyer making those three not guilty either.

So the culprit is the housekeeper.

This solution focus more on 'alibi'. Wife and Daughter alibied each other, as also Manservant and Lawyer pair, so all 4 must be innocent since only 1 is lying (no co-conspirators etc). That left the Son and Housekeeper under suspicion. But Son was seen along (I meant to say partially coexist; contemporaneous) with the people he alibied for by 2 other person, W who saw DS and L who saw SM. W's and L's statements could now be considered as ones which corroborate on S's statement, by extension, because they gave the confirmation that S did not and could not lie about seeing both D and M. So H must be the culprit.

Another way to look at this is that we have pairs who alibied each other; the WD pair and LM pair. As the criminal is working solo, then you can trust that neither members of these pairs were cahooting a murder plot together. That leaves us with the other kind of suspects, two of them in fact, who saw (self-claimed, of course) and being seen at the scene, but not mutually.

We can trust that L & W did saw S, but S claims to having seen D & M would require further investigation. Similarly, we can trust that M did saw H, but for H admitting to having seen W & L, a little digging would be necessary before we can believe on this claim.

It's obvious that while it wasn't D & M themselves who saw S, he was still witnessed by the two's other half of each pair (W of WD pair & L of LM pair). Even if the two pairs aren't contemporaneous to each other, but by the way that S was connected to both in this witness business, it can only mean one thing; that S gotta be the bridge between the two periods and S's time at the hospital room was overlapping both.

Now if we looked at H, only one of the pair did see H (this sole witness is M of LM pair) even though H claimed to have seen both W & L (one of each pair). Anyhow, it's quite believable for H to see L the way that H was being seen by L's pair, but with nothing to support H's claim on the other pair, WD, or at least the solo bridge S, it's seems far-fetched that H did see W as she claimed.

Further explanation for the above arguments against H claims to see W
==> W saw S who saw M who saw H who saw W. This is a 4-cycle.
==> There are 5 ways for this 4-cycle to actually truly happen :
1) All 4 of them were there at the same time,
OR
2) H saw W saw S happened first (around the same time), then W left, then M arrived, for him to be seen by S and he saw H (around the same time),
OR
3) W saw S saw M happened first (around the same time), then S left, then H arrived, for her to be seen by M and she saw W (around the same time),
OR
4) S saw M saw H happened first (around the same time), then M left, then W arrived, for her to be seen by H and she saw S (around the same time),
OR
5) M saw H saw W happened first (around the same time), then H left, then S arrived, for him to be seen by W and he saw M (around the same time).
==> Either way, all of the above scenarios implied that AT LEAST one of the two opposing vertices pair have to occupy the the same room at the same time together.
==> But H-S pair never mentioned each other the same way that W-M pair never met each other.
==> Thus, with the AT LEAST conditions cannot be fulfilled, then all 5 scenarios never happened at all and H claims to see W is rejected due to the impossibility of those circumstances.

This could be the sequence of the suspects arrival and departure time to and from the hospital room :

W --> D --> S --> L --> M --> H

1} The wife said she saw the daughter and the son. ==> W : DS (distance = 1 apart)
2] The daughter said she saw the wife and the lawyer. ==> D : WL (distance = 1 apart)
3) The son said he saw the daughter and the manservant. ==> S : DM (distance = 1 apart)
4] The manservant said he saw the housekeeper and the lawyer. ==> M : HL (distance = 0 apart, between the 2)
5) The housekeeper said she saw the wife and the lawyer. ==> H : WL (distance = 3 apart)
6} The lawyer said he saw the son and the manservant. ==> L : SM (distance = 0 apart, between the 2)

--[2{1-- W --(3-- D --{6-- S --1}[4-- L --2]-- M --3)6}-- H --4]--

Your situation could have happened, but then there would be MORE than 1 lie spoken. Daughter and housekeeper are saying the same thing, so if they BOTH really told the truth, then both should have been at the hospital at a time when witnessing both wife and lawyer is possible. Wife didn't see lawyer, AND lawyer didn't see wife. By the rules stated for this puzzle, if neither mentioned each other, then their time at the hospital didn't overlap. Wife left before lawyer arrived, or lawyer left before wife arrived, depending on who arrived there first. Repeating, if BOTH daughter AND housekeeper really told the truth, then both should have been at the hospital at a time when witnessing both wife and lawyer is possible. Then we found out that the witnessed wife and lawyer didn't even get to meet each other, so the supposed witnesses must have been there at the hospital AT LEAST BEFORE one of them leave and UNTIL AFTER the other arrive. Now let's look deeper into what was said. Wife and daughter witnessed each other, so the only way to reject the fact that they spent some time together at the hospital is for them to be BOTH lying, but this can't happen with the establishment of the one and only murderous liar. As for the lawyer, daughter claimed to see lawyer who saw son, who in turn saw daughter. In the same way, housekeeper claimed to see lawyer who saw manservant, who in turn saw housekeeper. Either could have lied by the same arguments, so let's see if housekeeper really saw wife. Because wife saw son who saw manservant who saw housekeeper who in turn, claimed to see wife, this cycle is much longer than the previous two 3-cycles. If this 4-cycle was the truth, then it was implied that this quadrilateral must have been made up by at least 2 'possibly' true 3-cycle triangles. Since wife and manservant didn't mention each other, we should have WSH + MSH triangles to be the supporting facts to housekeeper's WSMH claim. Unfortunately for her, she forgot to mention about seeing son even when she knew son wouldn't have any motives to lie and mentioned seeing her himself first, unprompted.

Let us consider a possible situation. Let the wife be the first to arrive . The daughter and son came soonafter . The wife saw daughter and son and left. Then came the lawyer. The daughter saw him and left. Then came manservant. Son saw him and left. Then housekeeper came.The lawyer, after seeing son and manservant earlier left. The manservant saw housekeeper. The housekeeper saw the lawyer earlier and there is no way he can see the wife. Thus he was lying. Hence he is the culprit.

Because housekeeper is seen by manservant only.and every other person is seen two times

The housekeeper said he saw the wife and the lawyer. But as we see, neither wife nor lawyer saw the housekeeper. So, he was lying.

Each person has to be seen by 2.

> Means "" seen by""

Daughter >> Wife and Son

Son >> Wife and Lawyer

Lawyer >> Daughter and Manservant

Manservant >> Lawyer and Son.

Housekeeper >> Manservant only. So Housekeeper lied. Notice that the Manservant did not lie.
It was the Housekeeper that lied because he was only seen by the Manservant. And yet he claimed to have seen the Lawyer, who was already seen twice.