Harry Potter's wand is a little unusual. It's a combination of phoenix feather and holly, which isn't often seen in the wizarding world. In addition, its core came from the tail of a phoenix called Fawkes, who was Albus Dumbledore's companion until his apparent death. Fawkes has donated only one other feather for use in making wands. That feather is in the wand belonging to Voldemort. This makes it difficult for Voldemort to attack Harry directly.
Voldemort has encountered numerous difficulties when trying to fight against Harry. The courage and ingenuity of his friends have often taken Voldemort by surprise. Harry's mother, Lily -- who used a wand made of willow wood -- gave her life to defend Harry against Voldemort. In the past, this sacrifice has protected Harry against his foe, although Voldemort's use of Harry's blood in his resurrection ritual seems to have dampened or destroyed this power.
But the greatest stumbling block seems to be the Priori Incantatem effect. When two wands both have cores from the same source, they will not function against each other. Instead, with enough focus on the wizard's part, one wand will overpower the other and force it to display all of the spells it has cast in reverse order. This is true only when the cores are from the same specific animal, not just the same species of animal. The precise cause of Priori Incantatem is not currently known.
Priori Incantatem is a rare event in the wizarding world, and it has some surprising effects. When two wands that share a core are forced to fight against each other, the winning wand may gain some of its opponent's attributes. In other words, in overcoming Voldemort's wand in a duel just after the Triwizard Tournament, Harry's wand absorbed some of Voldemort's power. The holly-and-phoenix-feather wand also gained the ability to recognize Voldemort as the owner of its brother wand -- even if Voldemort was using someone else's wand at the time. As a result, Harry's wand became able to channel extremely powerful magic when fighting Voldemort. It could even perform magic beyond Harry's knowledge or skill, without Harry's conscious effort.
This transfer of power could have made Harry's wand virtually unbeatable when fighting Voldemort. For this reason, Voldemort searched for, and discovered, a legendary wand known as the Elder Wand. This wand had the potential to be unbeatable in all duels, not just when fighting a specific opponent.
However, wands are fickle, and they follow their own particular rules when it comes to ownership. For reasons involving disarming and duels, the Elder Wand wasn't loyal to Voldemort. Wands are also relatively fragile -- Harry's wand was broken in another encounter with Voldemort, and repairing it was beyond even Mr. Ollivander's skill. Combined, both of these factors meant that Voldemort's final encounter with Harry did not go as he expected. Harry had a different wand, and the Elder Wand did not recognize Voldemort as its owner.
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