What does the phrase "a stick sharpened at both ends" mean in chapter 12 of Lord of the Flies?

It just means that Jack and the hunters are going to kill Ralph, cut off his head, and offer it as a sacrifice to the Beast (which is nothing but a dead paratrooper).  One sharp end goes into the ground, the other sharpened end will hold Ralph's head.  With Ralph gone, the island will be totally ruled by Jack and Roger and what little civilization is left will crumble.  Sam and Eric have been captured now and will have to do whatever they're told.

The reference in chapter 12 to the "stick sharpened at both ends" comes from the hushed conversation Ralph has with Samneric when Ralph is hiding from Jack and all the other boys.  Ralph is being hunted, just like a pig, and he is a tribe of one with all the others on Jack's side now, either by choice or by coersion.  Samneric tell Ralph that Roger sharpened a stick at both ends because Roger means to put one sharpened end into the ground and put Ralph's severed head on the other sharpened end of the stick.  Ralph would be regarded in the same way the boys regarded the pig whose head they earlier stuck on a stick that was sharpened at both ends.  That particular pig was treated that way as a sacrifice to the fictional "beast" that the boys in Jack's group feared.  That head became the Lord of the Flies that Simon, and later in chapter 12, seemingly spoke and identified the real evil on the island as the savagery inside each person.  By reducing Ralph to nothing more than a pig, the boys have reached complete savagery.

In chapter 12 of Lord of the Flies, what is the significance of Sam's statement, "Roger sharpened a stick at both ends"?

In chapter 8, when Jack and his group savagely kill the sow, Jack decides to offer a sacrifice to appease the beast, even though he is reluctant to openly admit there is a beast.  He orders the boys to sharpen a stick at both ends.  On one end, he plunges the sow's head and then he sticks the other end of the stick into the ground.  When, in chapter 12, Sam tells Ralph that Roger sharpened a stick at both ends, he is inferring that Ralph's head, once Ralph is caught will be plunged onto a stick and that stick will be put in the ground to display Ralph's head in the same way that the sow's head was displayed.  Ralph can't quite grasp the meaning of this because he still believes in civility and he is having a difficult time believing that the other boys want to hunt and kill him as if he were an animal.  He hasn't truly learned yet what the Lord of the Flies told Simon - that the evil is within each one of the them.

When and why does Roger sharpen a stick at both ends in Lord of the Flies?

In Chapter 8, the wilder boys that follow Jack have just violently and savagely killed a pig, and Jack tells Roger to sharpen a stick at both ends so that they can leave part of the pig's body as a sort of offering. Some of the boys believe that there is a beast alive in the jungle near their camps, and so Jack plans to leave the sow's head as a gift for the beast so that it will not attack the boys. Jack wants the stick sharp at both ends so that one end can be jammed into the earth and one rammed into the decapitated head of the sow they killed, so that the head stands up on the stick. Simon watches from his hiding place in the trees and first conceives that the head is the Lord of the Flies because of all the flies that are drawn to the meat and blood.

Roger is the one boy in the group that appears to have completely accepted his own evil and is completely willing to kill or hurt in order to accomplish whatever purpose he sees as important.  It is clear as he progresses from playing at being hurtful and cruel to looking forward to it.

By the time Piggy and Ralph come to the hunters to ask for the glasses, he has gone over to the dark side and relishes the chance to drop the huge rock and kill Piggy.

He sharpens the stick at both ends in chapter twelve, and it is Samneric who tell Ralph about it.  It isn't entirely clear what he intends to use it for but given their attitude it would appear to be some kind of torture device.

Why does the spear symbolize Jack in Lord of the Flies?

In William Golding's allegory, Lord of the Flies, Ralph and Piggy represent civilized man with the conch as symbol that calls the boys to meetings held with parliamentary proceedings while Piggy's glasses are symbolic of his high rationality. On the other hand, the leader of the boys' choir that originally have worn black robes and hats has descended to wearing painted masks and carrying spears.  The spear of Jack represents his savage nature that advocates beatings for the boys who do not comply with orders. In addition, the spear symbolizes the evil that man does when the controls of civilization are released. In Chapter Nine, for instance, when Simon crawls out of the forest, the hunters chant "Kill the beast! Cut his throat! Spill his blood!" and they circle around him as "the sticks fell and the mouth of the new circle crunched and screamed."

When the hunters do kill a pig, they cut its head and impale it on a stick, leaving it, as pagans might do, as an offering to the "beast."  It is this impaled pig's head that transforms into the "lord of the flies" that has spoken to Simon.  Later, when the hunters search for Ralph, they poke their spears through the bushes, hoping to stick Ralph.  Finally, the spear represents also Jack's fixation with hunting, a fixation that leads not only to his neglecting the rescue fire, but also to the bestial anarchy and destruction of the beautiful island.

What could be the implications of the phrase "a stick sharpened at both ends" in Golding's Lord of the Flies?

The stick sharpened at both ends definitely alludes to the fact that Jack and his tribe are planning to hunt down, kill, and behead Ralph just as they had done to the sow they offered as a sacrifice to the Beast.

Of course, neither the twins nor Ralph know at this point about the sow’s head. It isn’t until later, when Ralph encounters the meatless, horrifying skull atop a sharpened stick, that he makes the connection with what Samneric told him as he hid.

Therefore, the stick sharpened at both ends serves as a cryptic warning of the dehumanizing, savage violence that Jack wishes to exercise on Ralph. Interpreted this way, one could certainly argue that the stick represents pure evil; however, I would suggest that the stick is representative of violence while the Lord of the Flies itself (sow’s head when Simon “talks” to it) symbolizes evil.

Ralph is huddled in the jungle near Castle Rock, as dusk passes. Sam and Eric, who are guarding Castle Rock under Jack's leadership, are approached by Ralph. They inform him that Jack's tribe will be out hunting him the next day, and they have "sharpened a stick at both ends." The "stick sharpened at both ends" is a reference to what they did with the Lord of the Flies, impaling one end in the head and the other in the ground. This shows the brutality of what they planned to do to Ralph, who indeed felt like a pig being hunted by the savages.

Enotes offers an excellent analysis of the novel, and the second URL I provide below is a site completely dedicated to it.