# Flowers For a Patient

A woman wanted to visit her friend at the hospital. She had walked 500 meters from her front door before remembering that she wanted to pick up some flowers. She went back to the florist she had walked past, bought some flowers, and continued to the hospital. It was another 800 meters to the hospital. After visiting, she walked straight home using the same route.

How many total meters did she walk between leaving home and returning?

• How far did she walk from the front door of the hospital to her friend's bedside? :-) – Joe Oct 6 '14 at 20:44
• Yeah we seem to missing some information. Are you saying that there is information in there to provide a straight numerical answer? – d'alar'cop Oct 6 '14 at 20:50
• Yes, there is. Ignore the distance within the buildings, i.e. steps to the counter in the florist's, steps to the hospital bed, etc. – generalcrispy Oct 7 '14 at 13:03

Label her house $0$, the position of the florist $F$, the point where she stopped $S$, and the position of the hospital $H$.

She walked $S$, stopped, walked $S-F$, bought flowers, then walked $H-F$ to get to the hospital. Finally she walked $H$ to get home. That makes a total of

$S+(S-F)+(H-F)+H$ $=2S + 2H - 2F$

From the information in the question, we know that $S = 500$ and $H-F = 800$, so that gives

$1000 + 1600 = 2600$ total.

The apparently counterintuitive nature of the question is that it seems like we need to know how far she wasted retracing her steps between where she stopped and where the florist was. But note that the distance from the hospital to the florist is fixed. So the further she had to retrace her steps, the shorter we know her total journey was in the first place.

For example, if she only just passed the florist when she remembered, the total distance between house and hospital would be 1300m. If the florist was right next to her house and she had to trapse all the way back home to get those flowers, then the total distance between house and hospital would only be 800m.

So the distance between house and hospital decreases by exactly the wasted distance. The wasted distance is walked twice as a result of the waste (walking from the stopping point back to the florist, then getting back from the florist to the stoppoing point again), but the full journey is also walked twice, so the two even out.

• Did you know your answers are AWESOME? :-) Complete and fast. I really liked all of them. gratz and +1 – Rafe Oct 6 '14 at 20:57
• Yes, me too. You're awesome. BUT "The florist is in a straight line between her house and the hospital, 800 meters from it." - the function of "it" is ambiguous here – d'alar'cop Oct 6 '14 at 20:58
• @d'alar'cop Yes, I interpreted that as the hospital being 800 meters from her home, which renders the puzzle unsolvable. If his ambiguity is unintended, the question should be rephrased. – SQB Oct 7 '14 at 7:20
• Nicely done, Ben. – generalcrispy Oct 7 '14 at 13:04

@Ben Aaronson 's answer is the correct one, but I wanted to apply a technique called "The Zero Option" by Erwin Becker in "The Ultimate Book Of Puzzles, Mathematical Diversions, and Brainteasers".

(...) One telltale sign is an apparent lack of sufficient data, which implies that the solution is independent of the missing information. On this assumption the relevant dimension can be taken as zero. (...)

The puzzle seems difficult because we don't know the distance from the point where the woman remembered she should buy flowers and the florist. Per the Zero Option, if the puzzle has a solution, then this number must be irrelevant, and can be taken as zero.

So the woman walks (the equivalent of) 500 meters, then another 800 meters, then the same amount in the way back, for a round-trip total of $2\cdot(500+800) = 2600$ meters.

Of course, arriving at the correct result without understanding why it works is not as fun. But it can nudge us in the correct direction, which is writing the problem in a way that the unnecessary values disappear.

• You can convince yourself that it works by trying the case "The florist is opposite his home" (500 and back, then 800 and back) and the case "he remembered right at the florist" (500, then 800, and 500+800 back). If you get the same solution for both cases, it applies probably for all cases in between. – Florian F Oct 8 '14 at 19:37

Well, the florist is in between so, since it's 800 meters away, I'd say that the total distance from house and hospital is 1600 meters. So the florist is about 800 meters from her house. So that don't really alter her path, so she walked 1600 + 1600 meters, 2800 meters. Give or take a few for walking in the florist, and to the patients room.

That or I'm misreading/misunderstanding the question?

• Well, she walked 500 meters then went back to the florist, so I think from that the florist has to be no more than 500 meters from her house. – Ben Aaronson Oct 6 '14 at 21:02
• Yes, I thought it said somewhere in the question that it was midway. I thought catch was you didn't have to go back to the point. – warspyking Oct 6 '14 at 21:04
• Should I delete this? – warspyking Oct 6 '14 at 21:05