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Mordekaiser - Side Lane or Solo Mid and Picking a Lane Partner
Though generally melee heroes aren't particularly good at soloing mid, due to the high number of ranged harassing heroes vying for first blood, Mordekaiser works surprisingly well. His shield prevents him from being harassed (if you picked survability runes, you will be nigh unkillable to traditional mid heroes like Ashe and Teemo) and his spammable AoE abilities make him extremely capable when it comes to harassing back. If you don't have a dedicated solo mid, consider taking the position yourself - though understand that this is more difficult than laning. It may also be hard to keep your shield charged against very aggressive players or powerful long-range nukers like Anivia or Ezreal.
In a lane, try to end up with a fragile ranged carry or another pusher - paired with Sivir or even Soraka, Mordekaiser can be threatening the enemy tower at just a couple minutes in. In addition to this, your harassment ability means that enemy champions will need to stay behind their creeps, leaving your allied champion free to farm and harass at their convenience. Avoid pairing with other tanks or support heroes - your damage will generally not be high enough to compete with your opponents. If you plan on playing aggressively, lane with someone with a disable, since Mordekaiser lacks any of his own.
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Strategies for Early Game
Mordekaiser is one of the strongest early game champions. With a regeneration item like a Doran's Shield or Regrowth pendant, he can spam his AoEs and get easy creep kills without putting himself at risk. If you're in a side lane, feel free to be aggressive at the beginning - Mordekaiser has plenty of health already and with his shield he has an early game advantage. When creeps show up, use your Siphon of Destruction and Mace of Spades to last hit. Around level four or five, these two abilities will clear a creep wave if used at the right time. Try to catch opposing heroes in the AoE of each of these - since you're going to be using them anyway, so the harassment comes at no cost to you. Once you've worn down your opponents, don't be afraid to tower dive if you've got full shield, though be aware that it will only absorb one or two hits. Since you don't have a disable, you may be unable to finish off your enemies - this is ok. Use the opportunity to push the tower, though don't get overconfident - Mordekaiser can still be taken down relatively quickly early game, especially under tower fire.
Once you hit level six, try and coordinate with your lane mate or a ganking ally to kill someone in your lane using Children of the Grave, then use the ghost to take down the tower. Including the ghost, you could have three or four heroes pushing it, allowing you to do serious damage, taking it down or forcing a confrontation.
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Mid Game Strategies
Mordekaiser has two roles during the mid game - participating in teamfights and pushing towers. Since your abilities can clear creep waves easily and quickly, take as many opportunities as you can to give your creeps momentum and deal damage to undefended towers. Don't overextend, but also don't be afraid to push if your enemies don't have many disables. However, don't neglect team battles. Your allies will want you there. Use Creeping Death on your carry, since it will give them armor and magic resistance, and use your ultimate to designate an enemy to focus on - be sure to communicate to your team that this is your target. If you manage to kill an enemy with your ultimate, use the summoned spirit to turn the tide or push a lane.
When possible, kill the lizard and get the red buff. The slow will make you a better ganker and allow you to kill enemies solo. Once you reach level 11, you should again try to kill an enemy with Children of the Grave and use the spirit to push or force a confrontation - the combined power of an extra champion and the boost it gives to you is too good to pass up.
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Late/End Game Strategies
By end-game, regardless of how you've built Mordekaiser, you should be an indestructible juggernaut. Since Mordekaiser is more of a heavy melee character than a pure tank, much like Malphite or Nasus, let your initatior lead, then hit as many enemies as you can with your abilities to build up your shield and head straight for the casters and carries. Tough as you are, you will either kill one of them, force them disengage, or force your opponent's tanks to target you. Your team will be able to follow in your wake and finish up.Mordekaiser can survive a surprisingly long time in a team fight, especially if he has creeps and enemy heroes to charge his shield on. Don't be afraid to charge in if you're sure you've got support.
Keep pushing towers when appropriate, and use your AoE abilities to clear creep waves, giving momentum to your own creeps. If backdooring becomes necessary, Mordekaiser is more than capable of both fending off creep waves and tanking tower hits.