# Who Visited and When?

During a particular year, exactly ten people ($$A - J$$) visited a certain city on five different days between January 1st and April 30th, in a non-leap year; such that on each of the five days, exactly two of the ten people visited the city.

It is also known that:

1. $$A$$, who did not visit the city after $$J$$, visited the city $$28$$ days after $$F$$, who, in turn, did not visit the city with $$B$$.
2. $$J$$, who visited the city in March, visited the city at least $$50$$ days before $$C$$ visited but visited the city on the same day of the week as $$C$$.
3. $$D$$, who visited the city exactly $$10$$ days before $$H$$, visited the city with $$G$$.
4. Both $$E$$ and $$I$$ visited the city on February 10th, while $$E$$ and $$H$$ visited the city on the same day of the week.

On which day did $$B$$ visit the city?

Source : time.com

April 28th.

We can start by figuring out who visited the city with whom.

$$E$$ and $$I$$ visited the city together; so did $$D$$ and $$G$$. $$F$$ didn't visit with $$A$$, nor did they visit with $$B$$. $$J$$ did not visit with $$C$$. Since $$A$$ didn't visit after $$J$$ but did after $$F$$, $$F$$ and $$J$$ were not together. Further, $$A$$ visited $$28$$ days after $$F$$. Since $$C$$ is in April (see below), $$F$$ and $$C$$ aren't together. Therefore $$F$$ and $$H$$ are together. We either have $$A-C$$ and $$B-J$$ or $$A-J$$ and $$B-C$$. $$A-C$$ must come at least before $$B-J$$ because $$A$$ was at least before $$J$$; but it must come strictly after $$B-J$$ because $$J$$ precedes $$B$$. Contradiction! So we have our pairings: $$E-I$$, $$D-G$$, $$F-H$$, $$A-J$$, and $$B-C$$.

Now we'll start to make some assumptions or play around a bit.

Since $$J$$ visited in March and at least $$50$$ days before $$C$$ and on the same day of the week as $$C$$, we know that $$C$$ must have gone in April. Further, $$C$$ must have gone exactly $$56$$ days after $$J$$, since $$63 \hspace{0.25ex} days > 2 \hspace{0.25ex} months$$. Assume that $$C$$ went on April 30. Then $$J$$ must have visited on March 5. We therefore have a five day window for $$J$$ and $$C$$: $$J$$ between March 1 and March 5; and $$C$$ between April 26 and April 30.

Let's start to make hypotheses.

$$F$$ and $$H$$ being together means that they are on the same day of the week as $$E$$ and $$I$$, and also as $$A-J$$ and also therefore as $$C-B$$. $$D$$ and $$G$$ must be 10 days before that. So from above, we know that $$A-J$$ must be on the same day as February 10, between March 1 and March 5. Hence $$A-J$$ visited on March 3. Further, $$B-C$$ visited exactly 56 days after $$A-J$$, from above. We conclude that $$B$$ visited on April 28th. (To round out the answer, $$A-J$$ went exactly 28 days after $$F-H$$, so this would mean $$F-H$$ went on February 3rd. $$D-G$$ went 10 days before that, on January 24th.)

To summarize:

$$D-G$$ went on January 24th.
$$F-H$$ went on February 3rd.
$$E-I$$ went on February 10th.
$$A-J$$ went on March 3rd.
$$B-C$$ went on April 28th (which is the answer to this puzzle).

• Glorious answer as always, but I am just curious as to how you arrive the exact day count for $J$ to $C$? – PerpetualJ Oct 10 '18 at 15:49
• @PerpetualJ: You note that J and C visited on the same day of the week. So the day count is a multiple of 7. Then the difference has to be at least 50, so it's going to be 56, 63, 70, etc. But if J visited in March, then the maximum difference is between March 1 and April 30: a 60 day difference. So the day count is a multiple of 7 between 50 and 60 - hence it must be 56. – El-Guest Oct 10 '18 at 15:51
• Simple oversights; got to love them. :/ Good find! +1 – PerpetualJ Oct 10 '18 at 15:55
• You had the right idea and right way of thinking for sure! Good on you as well, @PerpetualJ! – El-Guest Oct 10 '18 at 15:58
• Glad to help @sam! Hope this is correct! – El-Guest Oct 11 '18 at 2:57

$$B$$ visited on:

Saturday, April 21st, 2018.

## Pairs and Dates

$$E$$ and $$I$$

Visited on Saturday, February 10th, 2018.

$$D$$ and $$G$$

Visited on Wednesday, January 24th, 2018.

$$J$$ and $$A$$

Visited on Saturday, March 3rd, 2018.

$$C$$ and $$B$$

Visited on Saturday, April 21st, 2018.

$$F$$ and $$H$$

Visited on Saturday, February 3rd, 2018.

## Reasoning

$$A$$ not after $$J$$ but $$28$$ days after $$F$$ and $$F$$ didn't visit with $$B$$.

This tells us that $$A$$ cannot visit after $$J$$ nor before $$28$$ days after $$F$$ has visited. Thus the smart thing to do is to pair $$A$$ with $$J$$ since this is not technically after $$J$$ (which threw me off as I was avoiding the pairing until I read the phrase more carefully).

$$J$$ in March; at least $$50$$ days prior to $$C$$ on the same day of the week.

This tells us that $$C$$ comes after $$J$$ and that $$J$$ is in March. This limits the available dates for $$J$$ and $$C$$ to land on. $$J$$ is limited to the 1st and 11th, while $$C$$ is limited to the 21st and 27th of April.

$$D$$ and $$G$$ visited together, exactly $$10$$ days prior to $$H$$.

This statement is pretty obvious. With the follow on statement regarding $$E$$ and $$I$$ we know that $$H$$ is on a Saturday, and thus $$D$$ and $$G$$ visit on a Wednesday 10 days prior.

$$E$$ and $$I$$ visited together on February 10th and $$H$$ visited on the same day of the week.

This gives us a fully taken date, which is a Saturday, and thus allows us to understand that $$H$$ also comes on a Saturday.

• You're so close, but I think your math for the 50 days might be slightly off...? – El-Guest Oct 10 '18 at 15:40