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You’ll get HM04 Strength from Grant, the Cyllage City Gym Leader. More specifically, you’ll get the HM on the Cyllage City bicycle track just before his gym. Cyllage City is located on the coast after the second half of Route 8 beyond Ambrette Town.
Note that you will not be able to use the move outside of battle until you’ve defeated Grant and earned the Cliff Badge, which is the second badge in the game.
Getting HM04 Strength
Step 1: Enter Cyllage City by heading north along the coastal portion of Route 8.
A swimmer on the beach will block your access to Cyllage City if you have not yet been to the Glittering Cave on Route 9 and the Fossil Lab in Ambrette Town.
Step 2: Follow the bicycle track (the tarred road) north past the Pokémon Center, and this will curve around as you go.
Step 3: The bicycle track curves east, behind the Cycle Shop and passing the Boutique.
Step 4: The track then turns south and starts heading up the cliff side, passing a cyclist who has come to a stop and the Cycle Shop again, now on the lower level to the west.
Step 5: The track will then curve back north to head further up the cliff side. As you take the turn it will trigger a cutscene where Grant will approach and tell you that you’ve already missed the bike race.
He offers you HM04 Strength as consolation for missing the race.
You cannot use HM04 Strength outside of battle until you have claimed the second gym badge from Grant himself.
The dirt path leading south off of the bicycle track goes to the Cyllage City Gym.
Using HM04 Strength
Once you have taught HM04 Strength to a member of your team, you will be able to use it to move big boulders around the map.
While Strength is not necessary to progress the game, it is useful to have in your arsenal as not only is it a powerful battle move but it also makes traversing the map easier and opens access to many secret areas.
Strength Boulder Puzzles
There are numerous puzzles on the map which can only be solved using Strength to move big boulders around. Once you push a boulder into a hole, you can then pass over it to enter new areas or create shortcuts.
A great example of this is the northern section of The Connecting Cave which connects Route 7 to Cyllage City.
This path is blocked by a big boulder when you first encounter it, and is thus not passable without the HM. Pushing the boulder into the hole creates a more direct way to move between the two locations. This route is useful for getting to and from the Daycare Center on or the Battle Chateau which are both on Route 7.
Note: Boulders do not usually reset. So once an area is opened it will stay that way.