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In the first Highlander movie, we see the Kurgan in a hotel room assembling his sword. It was a bunch of segments that fit neatly together like something you would buy at an IKEA for weapons (IKILLYA?). Even the blade came in multiple segments.

From what I understand, a blade needs to have a certain amount of give. Like a highway, it needs to be able to deal with a certain level of fluctuation and bending, otherwise it would just break when meeting substantial resistance. Now the reason why highways are not made of interlocking segments is because the bending and fluctuation would not all happen in the same direction and the segments would push against each other resulting in warping and deformation. We can expect the same problem from a sword that has a segmented blade.

I am aware of other segmented blades in fiction (Ivy's sword in SoulCalibur and Renji's Zanpakutto in Bleach), but those are less swords and more bladed whips. The Kurgan's sword operates purely as a standard bastard sword, so how does his blade not break on impact?

It would likely work badly. Skallagrim goes into it in more detail here:

The quillons are too thick, the blade is too heavy, the pop-out bits would likely break under anything heavier than a rapier, and indeed, the bits that snap together would likely fall apart with any real pressure.

That said, the Highlander wiki has an (unsourced as best I can tell) explanation, that the blade we see being assembled is a "practice blade":

The Kurgan fought with an emphasis on strength and brute striking-power, usually attempting to end the fight before it began with a single surprise attack. He would stalk his opponents methodically until he was ready. Then, he would attack without even so much as a challenge. He used a special two-handed broadsword, which he eventually supplemented with a detailed replica that could be disassembled and packed away as needed for practice in situations where the original could not be smuggled in and had to remain hidden, usually in his vehicle.

That the two blades are not seen together does not undermine the logic and resourcefulness of having a practice blade, that while not strong enough to last very long in combat, would give The Kurgan time to get to the original. It also recognizes the cunning and survival instincts of The Kurgan, adapting to the need for discretion in modern times while keeping his skills at their peak.

One of the key conceits in Highlander (ignoring wackiness we see in the sequel) is that the immortals have access to weapons technology that simply defies modern understanding of metallurgy; swords that are near-infinitely sharp, unbelievably durable and inconceivably resilient. Connor, for example has been using Ramirez' samurai sword to duel with for nearly 500 years (and he for 1500 years before that) yet it remains in defiantly mint condition when it should have been sharpened down to a nub.

JUAN RAMIREZ

climbing the trail to the forge. Olive skin, hawk nose, twinkling
eyes, flashing teeth. Flintlock pistol in his belt, crossbow across
his back.

Strapped to his side, a Samurai sword, carved hand-guard,
razor-sharp, feather-light.

_{^{Highlander: Final Draft Script}}

and

MACLEOD: This was forged in 593 B.C. Metal folded over 200 times.

He enjoys her amazement.

MACLEOD: (continuing gently) Like finding a 747 a thousand years before the Wright Brothers flew. Right?

_{^{Highlander: Final Draft Script}}

The Kurgan's sword is likely a product of the aforementioned weapons wizardry. It doesn't break simply because it's made of durable metals.

It's established that the immortals of Highlander have the ability to send their consciousness into the mind of an animal. It's not much of a stretch to believe that they put some of their life force into their swords, especially during a duel. That also explains the showers of sparks that often appear when their swords cross (although they needed cables and car batteries in the movies).

In a sense, then, the swords are alive, even magical, when wielded by an immortal. They carry a portion of the Prize. They heal their wounds. They cut through anything. They kill beings who cannot die.

I have always thought that Kurgans blade would have some sort of hollow locking mechanism that would extend a barbed rod from the hilt, further up the blade, and then lock into place pulling the 2 blade sections tightly together. It's the only way I could get my head around the thought that the little stub depicted would cause the top half of the sword spinning off to impale an innocent bystander 20 meters away the first time he crossed swords with anyone.

Even my purely imaginary mechanism still wouldn't hold up to actual combat. the weight and leverage on the joint make it impossible.

That just leaves The Rule of Cool or unsubstantiated theories. Maybe Candy was an Elvish sorceress in a mini skirt who repaired the broken blade in Kurgan's hotel room. It's as valid as any other theory.