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In the movie the Big Lebowski Walter explains that he is shomer shabbos and that he can't "roll" (meaning bowl) on shabbos. My question is, if he were able to get to the alley without violating shabbos and if he were allowed to bowl for free, other than possibly uv'din d'chol, is there any reason he can't bowl on shabbos?

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According to Maran (OH 308 IIRC) a ball would be Mukse (however the Rama is Mekil). – Hacham Gabriel Jun 26 '12 at 14:51
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The big question is, if Walter would go over the line and trip the sensor, would that be a psik reisha d'lo nicha lei? – Yserbius Jun 26 '12 at 19:56
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@Yserbius You need some chazara my friend - they didn't have sensors. If they did he wouldn't have argued with that guy over whether or not the guy went over the line. – Dov F Jun 26 '12 at 20:13
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@Yserbius this whole question may be moot considering Walter converted for reasons of marriage. But now we're getting into purim torah. – alltheinterwebs Jun 26 '12 at 20:22
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There may be Amirah L'Akum issues as well. Asking the owner to set up the lane, for example. Having a non-jew type in the info into the computer. – Menachem Jun 26 '12 at 20:27
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The rolling ball is the "koach" of the original person, analogous to a thrown rock. When the ball then hits the pins/sensors/etc that action is "koach kocho" of the original person, meaning it is twice removed from the person himself (for the definition see the Rambam Hilchos Rotzeach 6:15).

Now see the Avnei Nezer OH 230:2 who argues that "koach kocho" is potur on Shabbos, meaning that the act of the pin is only d'rabanan anyway. And the response from the sensor is psik reisha, as noted in the other answers. And the overall action of the machine is just an issur d'rabanan, since it is just electrical/mechanical activity that is not in the form of lighting/burning (as is clear from the tshuvos of R. Henkin and discussed elsewhere on this forum), no different from an electric fan.

This could perhaps be coupled with the view of the Rashba quoted in Shulchan Aruch OH 316 (Magen Avraham there) that allows psik reisha if the intention is a permissible act as well as an impermissible one (i.e. one wants to close the door, which is totally permissible, and in the process a deer is trapped inside).

So adding all this together, it seems clear there is sufficient room to be lenient about bowling on Shabbos, aside from issues of "uvda d'chol" which were excluded by the original questioner.

(As an aside, we often rely on this koach kocho argument for opening refrigerators on Shabbos, when the action of opening the fridge will let warm air in and surely turn on the cooling unit, a psik reisha that is nicha leih, but people are meikal; also perhaps because of the Rashba.)

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Are you sure about the koach kocho? It seems to me that the melacha that is being done through rolling the ball is still koach, unless you ar saying that the ball hits the pins and the pins trigger the sensors, but I'm pretty sure that the ball itself triggers a mechanism. – YDK Jun 26 '12 at 19:40
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@YDK I was under the impression that the mechanism is triggered behind the pins (in the "pin cushion block"), such that if pins are hit, then the pins do the triggering. You are right that if the person misses completely, then the ball would directly trigger the mechanics and it could be just "koach" – Curiouser Jun 26 '12 at 19:51
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@YDK but if he misses would it not be a psik reisha d'lo nicha lei? – alltheinterwebs Jun 26 '12 at 20:02
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@alltheinterwebs, only if he is a sore loser and doesn't want his score of 0 to count. Otherwise, he wants the mechanism to track his score, even if he doesn't like his score. – YDK Jun 26 '12 at 20:12
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Wikipedia addresses the question of permissability if only gutterballs are thrown thus voiding the questions of binyan and soser. I quote:

Of interest, this would include bowling, as the pins are knocked down, hopefully, during play. However, even if only gutterballs are thrown, the mere intention of knocking down pins is a violation of the prohibition against demolition. It is a matter of rabbinic debate as to whether intentionally throwing gutterballs during Shobbos is subject to the prohibition, as no demolition occurs. Commentary tends towards making this a violation as well, since the very act of bowling involves causing a machine to reconstruct the pin matrix upon each round, which action is initiated by the participating player. See further: Mishneh Torah Shabbos 10:15; Chayei Adam Shabbos 39, 43

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Chayei Adam and Rambam were bowlers?? – Seth J Feb 20 at 1:06
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In every bowling alley I've been to, the operation of the lanes depends on electro-mechanical devices that reset the pins and return the ball. As these actions are triggered by the bowler's bowling and are helpful to the bowler, they'd constitute a "pesik reisha denicha leih" - an expected consequence that's desirable - and would therefore be forbidden as use of electricity on Shabbat.

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since he is not activating those devices directly (but rather through the ball) would it not be a grama? – alltheinterwebs Jun 26 '12 at 15:59
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@alltheinterwebs, yes, and because the activation of the devices is a definite and desired outcome of the ball's arrival, it's in the subclass of grama called "pesik reisha denicha leih," which is forbidden. – Isaac Moses Jun 26 '12 at 16:16
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maybe i'm having trouble understanding the concept properly but I understood pesik reisha as a direct event. I cut off the head and it dies. Here by rolling a ball nothing happens immediately (in fact there is a distinct possibility that i miss the pins entirely!). wouldn't that be an indirect cause and hence only a grama? that coupled with the nature of electricity (possibly boneh?) would put the issur in doubt, no? – alltheinterwebs Jun 26 '12 at 16:20
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@alltheinterwebs, I may need to brush up on grama mechanics, but in this case, there's at least no uncertainty. Whether the ball hits the pins or not, it's going to activate the mechanisms that clear and return the ball. – Isaac Moses Jun 26 '12 at 17:20
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Nonelectric bowling alleys still exist. I was at one just a few years ago. We had to keep score by hand, so maybe gezeira atu koteiv? But it should be no worse than most card games. – Double AA♦ Jun 26 '12 at 19:09
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