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The Republic of Lateralia had been beset on all sides by hostile nation-states, each of which were eager to diminish the Republic's sovereignty. Lateralia's behemoth of an economy was an object of envy, and some ambitious kings and war-princes felt that the only way to become competitive would be to embark upon a campaign of subterfuge and sabotage.

Another thing that rival states couldn't stand about Lateralia was its plethora of high quality Chinese restaurants. Every city of appreciable size had a variety of take-out restaurants, all-you-can-eat buffets, and posh teahouses featuring a variety of food and drink based upon Asian cuisine. Agents in the employ of Lateralia's secret service had tailed two suspected enemy spies, codenamed Sherlock and Hercule, to a particular town. It was suspected at the time, and later confirmed, that Sherlock was somehow clandestinely communicating sensitive information about Lateralia's defenses to Hercule, but at the time, no evidence could be obtained to prove how it was done.

On August 29, Sherlock was observed dining at The Happy, Oh, So Happy Chinese Buffet. The eatery was known to be frequented by people suspected of anti-Lateral activities, and thus the entire establishment, with the exception of the bathrooms, were equipped with monitoring equipment such that authorized agents could record all of a target person's activity. Consequently, Lateral counterspies reviewed all the video and audio information, but unfortunately they were unable to find anything of particular interest, other than the fact that Sherlock ate quite a bit of broccoli and spent a fair amount of time in the bathroom. On August 30, the police arrived with a search warrant. Between the findings of their search and the sifting through of the trash from the previous day conducted by Lateral spies, no unusual physical objects were discovered as being left behind by Sherlock. On August 31, Hercule dined at the same eatery, and again his movements were carefully studied. It was noted that Hercule seemed to have a particular fondness for the doughnuts, and he also spent a fair amount of time in the same bathroom as Sherlock, but nothing could be determined about unusual behavior. Out of desperation, agents even detained and searched Hercule as he was about to drive away, but no papers or other communicatory instruments could be found on his person or in his vehicle. Nonetheless, Sherlock had successfully communicated the address and apartment number of a safehouse used by anti-Lateral spies.

The restaurant was generally restaurant-like, with only a few features of interest. There were stacks of plates, tables laden with food patrons could help themselves to by adding food to their plates with tongs, and mirrors along most of the walls. It was known that there was nothing hidden behind the mirrors other than the equipment Lateralia itself had installed for security purposes. There were eight individual bathrooms, four designated for males and four designated for females. Each bathroom communicated to the same hallway via a door with a padlock, which displayed an "occupied" message when locked and a "vacant" message when unlocked. Within each bathroom there were a sink with hot and cold water handles, a soap dispenser, a paper towel dispenser, a trash can, a mirror, a ceiling-mounted light, a motion-activated light switch, a toilet, a toilet tissue holder with four rolls of toilet tissue, a shower, and, for some strange reason, a dining chair. The four bathrooms for males also included a urinal "to help save water" (according to the owner). Also, each bathroom had a poster which read, "Employees must wash hands after using the bathroom. It would be appreciated if customers did so, too. -Management"

How was Sherlock able to communicate with Hercule without alerting the Lateral authorities?

Edit: For extra clarification, the following assumptions are reasonable:

  • The hallway that communicates with the bathrooms is not considered part of the bathrooms, and thus is under the same surveillance as the rest of the restaurant. Only the bathrooms proper are excluded.
  • The foreign spies visited one and only one bathroom during their visits. This happened to be the same one.
  • There were no secret compartments of any kind where papers or other conspicuous physical objects could be hidden.
  • There is some degree of maintenance and cleaning performed in the restaurant and each bathroom at the end of each day. Floors are swept and mopped; empty soap containers, paper towel dispensers, and toilet tissue rolls are filled or replaced.
  • Anything "left behind" by Sherlock would not be immediately obvious to the police or secret service under the conditions during which the restaurant was searched.
  • The address is short enough that it can be assumed it is easy to be memorized, but Hercule would only be able to learn it in the first place from his experience at the restaurant.
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    $\begingroup$ To emphasize, everything relevant occurred within the boundaries of the restaurant property. One could imagine that the "bubble" of the restaurant is within a Faraday cage, if you will, that somehow prevents the use of radio, mobile data, or communications equipment by spies to communicate outside the bounds of the bubble. $\endgroup$ – Wafflebaby Apr 20 '18 at 19:17
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    $\begingroup$ Can you confirm a few assumptions? 1) The hallway with the bathroom doors is monitored. Since Hercule used "the same bathroom" as Sherlock, I assume that if Sherlock had visited more than one of the 8 bathrooms, the SS (and thus the reader) would have noted it. 2) The SS is really good at searching, and would have noticed anything like items hidden behind ceiling tiles, vents, etc. Thus, the solution is not simply that Sherlock is better at hiding a piece of paper (/microchip/micro film/etc) than the SS is at finding it. 3) Restrooms are cleaned, toilet paper rolls changed, etc every day. $\endgroup$ – Selvek Apr 20 '18 at 19:24
  • $\begingroup$ @Selvek: I will edit the post. $\endgroup$ – Wafflebaby Apr 20 '18 at 19:25
  • $\begingroup$ Would the camera's be able to see if the light is on through the sides/bottom of the door? Also what kind of restaurant bathroom has a shower? $\endgroup$ – Shah.S Apr 20 '18 at 20:40
  • $\begingroup$ @Shah.S : I imagine that there probably would be some visible light at the bottom of the door, so the footage could probably indicate whether the light was on or not. This information by itself was insufficient for the secret service to determine what happened. As to the showers, that is a bit strange. It's possible the restaurant was originally intended as a front, or perhaps the owner likes to have slumber parties on days it is closed, and wants his guests to be able to wash themselves. The rationale is not particularly relevant. $\endgroup$ – Wafflebaby Apr 20 '18 at 20:52
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Could Sherlock have:

left a hidden message in the mirror using the soap. He writes the message using soap and cleans it up afterwards carefully. Also this doesn't have to be just the mirror it could have been the toilet, the urinal, and maybe even the poster

So when Hercule went in there

He just turned on the shower and waited for the steam to fog the mirror causing the message to show up

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  • $\begingroup$ This comment is likely to contain spoilers. Be warned! This is what I was thinking. To clarify for others, a possible way to leave a difficult to detect message would be to use a lipid or hydrophobic substance, such as fats used in soap, to form a message on the mirror. Since Sherlock spent a fair amount of time, he could have written a message and wiped off the excess soap. If the message was not particularly disturbed, Hercule could come later and use the hot water from the sink and shower to provide water vapor that would leave condensation on the mirror except where the message was written $\endgroup$ – Wafflebaby Apr 20 '18 at 22:41
  • $\begingroup$ Some of my research for this puzzle was based on the following sources: dougintology.blogspot.com/2010/03/… ; reddit.com/r/askscience/comments/3twio2/… ; tvtropes.org/pmwiki/pmwiki.php/Main/CondensationClue All credit for the idea belong to the respective authors. $\endgroup$ – Wafflebaby Apr 20 '18 at 22:43
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Here is one possible solution. Like any such solution, it is dependent on how thoroughly the SS searched the premises, and how good they are at recognizing hidden information, as well as how thorough the cleaning crew is.

The information was hidden behind the "Please wash hands" poster. Since this is a "high quality" chinese restaurant, we assume the cleaning crew is pretty thorough, and does things like wiping down the walls and mirror. However, it is unlikely that they would remove the poster. Now, lets assume that one of the broccoli offerings at the restaurant includes some kind of slightly sticky sauce - perhaps one containing some orange or lemon. Even a little bit of sugar would create a slight sticky spot on the wall. So Hercule need only grab a couple powdered sugar donuts, blow the sugar onto the wall, and see where it sticks.

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  • $\begingroup$ I was trying to come up with a solution that involved the sugar in some way. Although I doubt the restaurant does a thorough cleaning of the walls and mirror every single day. They probably rub them with a paper towel or something of that sort, but it seems to me that this might work on the walls as well. I would have to actually test this out to see how practical this would be. (How much sugar would be needed, the logistics of getting enough doughnuts/sugar into the bathroom without arousing suspicion, etc.) However, unless a "better" answer is posted, I think this is eligible for acceptance. $\endgroup$ – Wafflebaby Apr 20 '18 at 20:03
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    $\begingroup$ I can tell you from experience it takes very little sugar to make an annoying sticky spot on my kitchen wall :) Although a wet paper towel makes short work of it. I'm curious what your intended solution is. $\endgroup$ – Selvek Apr 20 '18 at 20:06
  • $\begingroup$ That's possible, but if the address is something like "12 Main St Apt 8", that would seem to require a bit of substance to get all the contours to be distinguishable. $\endgroup$ – Wafflebaby Apr 20 '18 at 20:07
  • $\begingroup$ Wouldn't it be suspicious if they were to grab food or take it with them. It says above that tongs were used by customers to take food, so unless they were really messy eaters they wouldn't have had enough residue on their hands for a long address. $\endgroup$ – Shah.S Apr 20 '18 at 20:56
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    $\begingroup$ Actually... for that matter, Sherlock could have carried in any kind of invisible ink in his pocket (he was never searched). Hercule could have brought in whatever item/chemical he needed to reveal the ink. As long as he flushed the item, and/or the packaging he carried the chemical in was something that wouldn't arose suspicion (e.g. a soda bottle or anything else that is commonly thrown out), he would have been fine since he wasn't searched until after he left. $\endgroup$ – Selvek Apr 20 '18 at 22:25
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Partial answer:

Broccoli contains phosphorus. Phosphorus glows in UV light (black light). He entered the bathroom with a mouthful of broccoli and wrote the address with it on the mirror. Now how do you make a black light out of doughnuts? No clue.

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  • $\begingroup$ I like this answer, and it is in line with my thinking when I was trying to consider alternate solutions. It seems to me as though it would be unlikely for the police to use a black light when they were ostensibly looking for papers and effects, so that might be a good way to hide the message. The hard part would be how to reveal it without being seen on the footage with strange apparatus, nor with it being found among Hercule's effects. $\endgroup$ – Wafflebaby Apr 20 '18 at 19:53
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    $\begingroup$ It also made sense to me that they would have to be in there a long time to get the light to turn off by standing still... $\endgroup$ – MetaZen Apr 20 '18 at 20:20

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