I know this is weird of me, but I always eat the top first.  The reason is I don't want to turn it upside down (sticky side up) to set it down while I eat the bottom half.  Possibly because I'm under the impression it's somehow more round than the bottom.
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I always make mine into a sandwich of sorts, so both halves are eaten at the same time.
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Which half of the bagel I eat first isn't intrinsic to the characteristics of the top or bottom half of the bagel, but rather is dependent on what is on each half of the bagel.

See I like to have a lot of variety in what I eat and so rarely will I have the same spread on both the top and bottom half. Whether different types of cream cheese, tuna, egg salad, or whitefish I always mix it up. Depending on the topping it can go on either the top or bottom depending on how strong its flavor is and what texture it has. Whitefish salad for example is better suited to the bottom half and chive cream cheese to the top.

With all that said I can now answer which half I eat first. I don't want to have the fish taste in my mouth last and so I will eat that half of the bagel (usually bottom) first.
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The Outer Half

I had to really think about this since there have been several bagel shops here in the past like New York Bagel, etc...There was definitely a strong bagel trend for awhile. I still see them at a few trendy shops like St. Louis Bread. With all of the different kinds of bagels, I had to think, was it a sandwich, bottom half, etc, like the answers I read above, then I thought, no, not really. The real truth is that we eat the bagels on the outside half. 

Discovering how we eat bagels, is really not that bad of a question. Like does this grant more thought that we could dip donuts and bagels on the outer rim, like tire treads of chocolate or nuts, etc, to promote a new style of bagel and donut. And also, are there such things as Bagel Holes? If not, that would be a really popular invention I think. Maybe if some of these bagel shops would be more creative, they would give Krispy Kreme a run for their money. I know that we all like something new, no matter what the correct answer to the question is...

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The whole thing all at once! Enjoy all the flavor and texture in a few quick bites.
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I eat the top first, always. I just love the crispier/thicker bottom of the bagel more than the top so I save it for last.

I have family who like the top more so sometimes you'll find a bag of bagels with all the top slices missing in my household - and the bottom slices waiting to be eaten by me.

om nom nom
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The top half with the toppings. I like the extra flavors.
Source(s):
Self

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The bottom half first. I usually get a cheese bagel with regular cream cheese... And I eat the bottom first, that way I can have the "good part" of the bagel last... Save the best for last..
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it is the same as a muffin, I eat the bottom first, rather quickly then I can savor the flavor of the top.  The top has a lot more flavor and way better texture for the reason that it sits in the oven getting blasted with all the flavor rendering heat and bakes up a nice crispy crunchy top
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I am a bit different, I never cut mine at all. I also do not toast mine either but eat the entire thing at once, with only butter.
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I toast mine in a toaster oven, and the put cream cheese on both halves.  Whichever one is the "toastiest" is the half I eat first.  When it comes to bagel halves, I don't like to save the best for last. 
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I have that question answered for me by my wife. Every time we go out for coffee and I say, "You want anything to eat?" she inevitably says no, and then eats half of mine. Luckily though she tends to eat the bottom, smaller half, leaving me the good one.
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Generally the top, particularly if it's a salt or "everything" bagel.

Eating a bagel is already indulging in a big dose of straight carbohydrates, so may as well make it a flavorful indulgence.
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Whichever half ended up being cut thinner. Better bread to cream cheese ratio. ;)
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I like to eat the center first becasue there are less calories. I've been on a diet since the holidays.
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I eat the top first, cut in half itself and untoasted as a small sandwich with lox and cream cheese or salad vegetables and cream cheese.  Then I toast the bottom which is a bit flatter, seems to toast more evenly, and has less coatings to drop into the toaster (I love everything bagels with onion flakes, poppy seeds, sesame seeds on them).  I eat the toasted bottom half with butter or cut inhalf with fillings as a small toasted sandwich.

 
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