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I read on Wikipedia

Various leaflets were dropped on Japan, three versions showing the names of 11 or 12 Japanese cities targeted for destruction by firebombing. The other side contained text stating "...we cannot promise that only these cities will be among those attacked.

In general, the Japanese regarded the leaflet messages as truthful, with many Japanese choosing to leave major cities.

To what extent did the requests/recommendations from the Americans to the civilians of Hiroshima and Nagasaki to leave the city help reduce the number of civilian casualties of the two nuclear bombs in August 1945?

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    Your link suggests Hiroshima was not listed among the cities and may or may not have been leafleted. While there may be examples of people who left Hiroshima for rural areas, there may have been others who went to Hiroshima. And then there was Tsutomu Yamaguchi who went to Hiroshima, got bombed, returned home wounded to Nagasaki and was bombed again. – Henry Apr 21 at 17:20
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    @Henry thanks yes, good point, this definitely makes the situation less obvious. – Franck Dernoncourt Apr 21 at 17:23
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    such questions are very difficult to answer scientifically. We would require parallel realities, in some of which the leaflets were dropped, in others they weren't dropped, and then compare the results... – vsz Apr 23 at 8:54
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There is a 1946 book by John Hersey, Hiroshima, which is an excellent compilation of personal testimonies from Hiroshima residents following the atomic bombing. Although it doesn’t mention Allied leafleting specifically, there’s a few important things it indicates about civilian perceptions of the threat at the time, namely

  1. It was abundantly clear to Hiroshima residents at the time, from the lack of conventional American firebombing, that the Americans were planning something terrible and that it was best to stay away from the city if possible. For example, city doctors were already turning away patients since they didn’t want to be responsible for evacuating them during what they expected to be a massive imminent fire raid. According to Hersey, at least 135,000 of the city’s ~380,000 residents had already evacuated by the time August 6th rolled around.

  2. Nobody (who wasn’t a government nuclear physicist) was expecting an atomic attack, since nobody knew what an atom bomb even was. Everyone was expecting just a scaled up firebombing. The Hiroshima municipal government had been preparing for this for some time (since they, like everyone else, were expecting an attack), to the point of drafting local teenage girls to clear fire lanes to nearby rivers. So the type of person who would be unfazed by threats — that is, most people still in the city, since the jumpier people had already evacuated — felt confident that the city could withstand a large American attack.

  3. Since July at least, American pilots had been using Lake Biwa (just northeast of Hiroshima) as a rendezvous point, so there were basically always American bombers in the sky, which meant that Hiroshima residents were always hearing air raid sirens. Since the bombers were simply flying over the city on their way to bomb some other city, this lead to a great deal of alarm fatigue among city residents. The night before the US dropped its atomic bomb, the city’s air raid warning system detected about 200 B-29s approaching the city, which forced a lot of grumpy people to get up in the middle of the night and run to shelter. It was a false alarm, so naturally even fewer people than usual were in the mood to take cover when the real bomb came the next morning.

  4. Since everyone was expecting a scaled-up firebombing, they assumed that any such attack would have to involve many many bombers. The air raid sirens went off any time they spotted at least one plane, but city residents had learned to distinguish between “harmless” reconnaissance flights and “dangerous” large formations of bombers. It also didn’t help that the Enola Gay came around the same time that an American weather plane (which was known to city residents, since it triggered the alarm at the same time every morning) made its regularly-scheduled pass.

To summarize, most people already knew some sort of attack was coming, but they probably wouldn’t have understood what an atomic attack meant, even if you explained it to them. They had also been hearing false warning after false warning, so those who had decided to stay probably wouldn’t have believed it anyway.

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It doesn't seem that either city knew with certainty that it was going to be subject to attack (nuclear or otherwise) on those specific days. Japan in general was experiencing relentless bombing. There were at least some evacuations in both cities, but these were not necessarily the result of the leaflets per se.

About Hiroshima, a Wikipedia page (the source it cites for this is behind a paywall) says that:

120,000 of Hiroshima's population of 365,000... evacuated the city before the atomic bomb attack on it in August 1945.

However, this article says that:

The bomb, dropped by the US on August 6 1945, made orphans of around 2,000 children, mostly from central Hiroshima, who survived because they had been evacuated to the countryside. [...] More than 90% of the population of central Hiroshima perished.

The answer to this related question on Reddit states (without providing sources) that:

A large group of evacuees were in fact gathering in the center of [Hiroshima] on the morning of August 6th, when the bombing happened. Bad timing.

About Nagasaki, this article from the US Dept. of Energy states:

A small conventional raid on Nagasaki on August 1st had resulted in a partial evacuation of the city, especially of school children. There were still almost 200,000 people in the city below the bomb when it exploded.

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    Reddit is not a reliable source. – Jurp Apr 21 at 23:33
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    I am sorry, I downvoted since you are not answering to the OP's question directly and providing the circumference information. This is not to "protect" my answer, but just I thought your answer is poor. User @taylor swift's is better. – Kentaro Apr 22 at 5:00
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    @Kentaro "circumference information"? This seems like a pretty straightforward answer to the OP's question to me. – hobbs Apr 22 at 21:17
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    @Kentaro The criteria is "This answer is useful." It doesn't have to be best or even excellent. Just useful. Downvoting answers with useful content in them misleads seekers. – Russell McMahon Apr 23 at 5:25
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    @RussellMcMahon: Up-voting answers with grossly misleading statements in them is worse. – Pieter Geerkens Apr 23 at 7:24

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