PokéBase - Pokémon Q&A
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Most species caught have 70 happiness (with the exception of certain species and N's Pokemon etc.).

Soothe Bell:
>holding a Soothe Bell will apply a multiplier of 1.5 (rounding down) to only positive changes and that use of Luxury Balls will not reduce the effects of negative changes.


Luxury Ball:
>A luxury ball doubles the happiness value of the Pokemon when caught. So normally, a Pokemon is caught with 70(depending on species) happiness but if you caught it with a Luxury ball, it would be caught at 140 happiness.


Walking 128 steps (50%)
0-99 +2
100-199 +2
200-Max +1

>Several Pokémon evolve after being leveled up when their friendship level is at or above 220


Umbreon/ Espeon being one of them. So 220 is the target.

When Eevee is caught in a Luxury Ball, its base happiness will be 140.

It will take another 59 before it reaches 199 happiness, and the steps required for that are:
(128 steps (+1) x 59) / 1.5 (Soothe Bell) / 2 (Luxury Ball) = 2,518 steps (rounded up)

(128 steps (+1) x 21) / 1.5 (Soothe Bell) / 2 (Luxury Ball) = 896 steps

Total: 2,518 + 896 = 3,414 steps.

This is only for walking, and in Gen V.

NB: can someone please double check my maths.

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"(128 steps (+1) x 59) / 1.5 (Soothe Bell) / 2 (Luxury Ball) = 2,518 steps (rounded up)"
I'm pretty sure that 128 * 59 / 1.5 / 2 = 2517,333 => 2517, no ? Or in Poké-Logic, we always round up to the higher value ? Because if we round up to the exact value, it may give 2517, not 2518.

I double-checked your maths, everything's fine, Calculator ! I didn't expected a fondant being so much calculator than a legendary dog (me ; Arcanine) ...
Thank you for that; also it said the multiplier of 1.5 was rounded down, but I used it in division, so I thought it would be rounded up to a higher number of steps?