A 21-speed bike only has 9 speeds

On a 21-speed bike, it is almost impossible to shift from smallest gear to largest gear in order. To do so, you'd have to shift something like the following example sequence. This would require shifting both the front and rear derailleurs at the same time, and jumping multiple sprockets at a single time like this:

First number is front derailleur, and second number is rear derailleur.

1-1
1-2
1-3
2-1
1-4
2-2
3-1
1-5
2-3
3-2
1-6
2-4
3-3
1-7
2-5
3-4
2-6
3-5
2-7
3-6
3-7

Anyone see a problem? Like, can you actually remember to go from gear combination 3-1 to combination 1-5 when cycling?? Based on this sequence, if you examined the actual number of practical gears available to you (gears you will actually use, where you strictly upshift a single gear at a time, without having to downshift another gear), a 21-speed bike really only gives you up to 9 "usable" linear gears:

Example 1:

1-1
1-2
1-3
2-3
3-3
3-4
3-5
3-6
3-7

Example 2:

1-1
1-2
2-2
2-3
2-4
2-5
3-5
3-6
3-7

Example 3:

1-1
2-1
2-2
2-3
2-4
2-5
3-5
3-6
3-7

This result is the product of basic math, as you've got 6 gear shifts on the rear derailleur, and 2 gear shifts on the front. This makes 8. Add 1 for your starting position, and you've got 9 usable gears.

So the bottom line is that on a 21-speed bike, there are 12 gears that you will likely almost never use (or need).

Now that you know that a 21-speed bike is really a 9-speed bike, a 7-speed integrated hub shifter doesn't sound so bad, does it?

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