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Earlier today I answered a question about the badge multiplier mis-alignment on the user profile page. While debugging the answer for that question, I went through few of the top ranked users profile pages such as Shog9, Oded, etc. where the badge count (x multiplier) was not shown for the top 3 Gold badges they have earned. But then, upon looking Tim Post and Monica Cellio's profile, I noticed the badge count is displayed next to some of their top 3 Gold badges.

Here, I would like to give an example of Shog9 and Monica Cellio's profile page for the comparison purpose. Pay attention to the list of Rarest badges under the Gold Badges section.

Here is Shog9's Badges section screenshot:

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and, here is Monica Cellio's Badges section screenshot:

enter image description here

I would like to know what kinda algorithm is used while determining when to display or not to display the badge count multiplier. Can anyone please explain this.

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The multiplier is only shown when the badge has been awarded multiple times, and that's really all there is to it.

In a majority of the cases in your examples, the featured badges can only ever be awarded once anyway, so there was never a chance of them being awarded multiple times, but this holds true even for multiple-award badges only received once:

my "rarest" gold badges

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