seat reservation needed with Shinkansen?

According to the hyperdia train information, is the charge for reserved seat compulsory on the shinkansen or other trains?

If I travel from Tokyo to Kyoto, Kyoto to Hakone then back to Tokyo, is the 7-day JR pass worth it? Should I get the Hakone free pass from Tokyo even if I use it one way?

According to the hyperdia train information, is the charge for reserved seat compulsory on the shinkansen or other trains?

Yes, it is compulsory if you travel by reserved seat (unless you have a Japan Rail Pass). The "unreserved seat" charge will apply if you travel by unreserved seat. The difference in price is typically 300-700 yen.

Most trains have some cars with unreserved seats, but some, like the Narita Express, have only reserved seats.

If I travel from Tokyo to Kyoto, Kyoto to Hakone then back to Tokyo, is the 7-day JR pass worth it?

Yes, if a round trip Tokyo-Kyoto-Tokyo fits into a 7-day period, a 7-day Japan Rail Pass will pay off.

Should I get the Hakone free pass from Tokyo even if I use it one way?

No, just get the Hakone Free Pass at Odawara, and ise the Japan Rail Pass between Odawara and Tokyo.

I don't have to take the reserved seat to save some $. But do I have a choice? e.g. from Shinagawa to Kyoto the reserved seat charge is 5740 yen extra.

But do I have a choice?

Yes, you have the choice between reserved seat (5740 yen by nozomi or 5440 yen by hikari during peak seasons) or unreserved seat (4730 yen). And if you have a Japan Rail Pass, no surcharge has to be paid, but you are not able to ride the nozomi.

: If I travel from Tokyo to Kyoto, Kyoto to Hakone then back to Tokyo, is the 7-day JR pass worth it?

The JR part will be Tokyo - Kyoto - Odawara - Tokyo.
Compared to 28,300 yen:
- If "Tokyo" is Tokyo Station, regular tickets cost lower.
- If "Tokyo" is a station in Narita Airport and you take a N'EX [Narita Express] on the first half with a costless N'EX ticket #, then regular tickets and the N'EX ticket
cost about the same when you don't take a N'EX on the second half,
cost higher when you take a N'EX on the second half.
# a costless N'EX ticket = a N'EX ticket in a Suica & N'EX package.
(These simulations are on the supposition that you will not take a Shinkansen from Odawara.)

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