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As I understand it, "Chacham harazim" is the bracha we make upon seeing an assembly of more than 600,000 Jews.

I'm not clear on the exact detail of this law/tradition, but I also know we are not supposed to count Jews.

Bracketing, for a second, the fact that one probably couldn't count a crowd of 600,000...How could one be expected to assess the conditions for this bracha without counting Jews?

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The Halacha is that you are not allowed to count people. But you could count noses, or shirts, or left shoes, or ... (source: Magen Avraham O.C. 156 - 4th wide line of right column here).

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See this article which related this question to the Atzeres Hatefila event. (I'm unfamiliar with what this event is or when it occurred. I just noticed the article as it relates to answering your question.) Excerpt:

Prior to the event, the question was raised whether the bracha can be recited if unable to ascertain whether there truly are 600,000 in attendance. Rav Yitzchok Zilberstein shlit”a came up with a brilliant solution: to read the halacha out of the Shulchan Aruch, as it contains the entire nusach of the bracha! That way even if the full amount was not present, in no way would one have possibly recited a bracha levattalah

See the footnote to the last point as well as the remainder of the article. It also mentions that there are some opinions stating that the bracha should be said only in Eretz Yisra'el.

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    What's so brilliant about that? It's the oldest trick in the book. – Double AA Nov 13 '14 at 21:06
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    IAE this doesn't answer the question... – Double AA Nov 13 '14 at 21:08
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    @SAH What, you don't already have a full library on your phone? (website version/iOS/Android) – Scimonster Nov 14 '14 at 11:37
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    @SAH Or, you could memorize the relevant Siman (or maybe just the relevant Se'if, or add a few surrounding Se'ifim) – Tamir Evan Nov 14 '14 at 11:40
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    @Scimonster He would, only if he followed those Rabanim who permit using such a phone, in the first place (to say nothing about permitting him to be connected to the Internet). – Tamir Evan Nov 14 '14 at 11:44

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