Why did Bhima kill Duryodhana unfairly by hitting him below the waist during their final battle in the Mahabharata war?
Yudhisthira actually deeply regretted the deceitful actions. He said in Mahabharata Salya Parva Section 59,
"It is we that deserve to be pitied in every respect, O Kaurava! We shall have to drag on a miserable existence, bereft of all our dear friends and kinsmen. Alas, how shall I behold the widows, overwhelmed with grief and deprived of their senses by sorrow, of my brothers and sons and grandsons! ..... We, on the other hand, shall be reckoned as creatures of hell, and shall continue to suffer the most poignant grief! The grief-afflicted wives of Dhritarashtra's sons and grandsons, those widows crushed with sorrow, will without doubt, curse us all!"
Krishna defended the various deceitful actions as follows in a voice deep as that of the clouds or the drum in Mahabharata Salya Parva Section 61:
"All of them were great car-warriors and exceedingly quick in the use of weapons! If ye had put forth all your prowess even then ye could never have slain them in battle by fighting fairly! King Duryodhana also could never be slain in a fair encounter! The same is the case with all those mighty car-warriors headed by Bhishma! From desire of doing good to you, I repeatedly applied my powers of illusion and caused them to be slain by diverse means in battle. If I had not adopted such deceitful ways in battle, victory would never have been yours, nor kingdom, nor wealth! These four were very high-souled warriors and regarded as Atirathas in the world. The very Regents of the Earth could not slay them in fair fight. Similarly, the son of Dhritarasthra, though fatigued when armed with mace, could not be slain in a fair fight by Yama himself armed with his bludgeon! Ye should not take it to heart that this foe of yours hath been slain deceitfully. When the number of one's foes becomes great, then destruction should be effected by contrivances and means. The gods themselves, in slaying the Asuras, have trod the same way. That way, therefore, that had been trod by the gods, may be trod by all."
Krishna argues that Pandavas could not have defeated the Kauravas in a fair fight. A defeat of the Pandavas would have meant defeat of dharma. Hence the deceitful means.....
Let’s first consider how unfair the whole battle was for Bhima. Due to Gandhari’s blessing Duryodhana’s body had become invincible, no matter how expertly or forcefully Bhima hit it. What was he supposed to do? Just let himself get beaten to death for no fault of his – all because his opponent had got a blessing for no virtue of his?
A “blessed” batsman
To grasp the unfairness of the situation, consider a rough cricket analogue: during a faceoff between a champion batsman and a champion bowler, suppose the batsman gets a “blessing” to never get out, even if he is caught, trapped leg before wicket, clean bowled, hit wicket or whatever else. What is the bowler supposed to do? No matter how well he bowls, there’s just no way he can win. Fans with even a modest sense of fairness would be up in arms protesting the way the contest had been rigged against the bowler. But suppose no one protested. The poor bowler bowled the best spell of his life, got the batsman out several times and yet got no credit for it. Wouldn’t that be patently unfair?
That’s what happened to Bhima. He hit Duryodhana twice with such awesome force that the blow could have rent a mountain apart, what to speak of breaking a human frame apart. His achievement was like that of a bowler clean-bowling the batsman not just once but scores of times. And yet what did Bhima get for his feat? Nothing – leave alone victory, not even a crack on Duryodhana’s body.
How long could Bhima go on like this especially when Duryodhana was counterattacking and wounding Bhima? Among the many blows that hit Bhima, two of Duryodhana’s blows were so brutal that they would have instantaneously killed a lesser warrior. Though Bhima was badly injured by those devastating blows, he with superhuman fighting spirit maintained a stoic face, showing no weaknesses. His plight was like that of a bowler carted for six sixes in two successive overs. Actually, Bhima’s plight was a million times worse. Why? Because Duryodhana’s blows were wounding not just Bhima’s morale, but also his body. It’s something akin to the batsman’s shots hitting the bowler, thus rendering him less and less capable of bowling – while still being expected to go on bowling till death.
Can we really blame the battered bowler if he took the only way out of the carnage: bowl bodyline and get rid of the batsman, retired hurt? If we were being wounded like that, can we be sure that we too wouldn’t do something similar?
Understandably, Bhima took the only way out of the slaughter by hitting Duryodhana at the only place it hurt: his thighs. Just as bowling bodyline in normal cricket is unfair, so was hitting the thighs unfair in a normal mace-fighting battle. But when it was the only way to bring some fairness back into an unfair battle, would it still be blameworthy?
Honoring the spirit of the rule, adapting its letter
Rather than blaming Bhima for hitting that unfair blow, perhaps we need to give him credit for fighting fairly for so long, despite being sentenced to an eminently unfair contest. Bhima could have claimed justification for finishing the battle quickly with an early unfair blow:
Maitreya Rishi had cursed Duryodhana that he would die due to the breaking of his thighs and Bhima could have claimed to simply be an instrument for fulfilling the sage’s curse.
Bhima could even have claimed that he had vowed to break Duryodhana’s thighs for having obscenely exposed those thighs to publically humiliate Draupadi – and that he had to do whatever it took to fulfill his vow.
That Bhima did not take recourse of any of these reasons at the first possible opportunity demonstrates his respect for for the spirit of fair play. But the battle had been rendered unfair not because of his action, but even before he took any action.
Their battle was not like a normal mace-fighting battle to which the standard rules applied. Duryodhana had brought something extraneous into the battle – his mother’s protection in the form of an invisible invincible armor. That extraneous factor so totally tilted the scales in Duryodhana’s favor that to re-balance the scales Bhima too had to bring something extraneous – Krishna’s protection in the form of an action plan to bypass that armor. Krishna implemented part A of that emergency plan when he persuaded Duryodhana to cover his private parts while going to see his mother. And then he told Bhima to implement part B of that plan by hitting Duryodhana’s vulnerable thighs.
Yes, that kind of blow was unfair in a normal wrestling battle, but what was normal or fair about a wrestling battle in which one player couldn’t win, no matter how well he fought? Understandably, such an abnormal battle couldn’t be played by the normal rules if there was to be any hope for a fair result.
As Krishna later said, there was no other way Bhima could have won – and so he had to take the only way available.
Duryodhana's upper body was made of vajras (diamonds) and hence no one can kill him by striking anywhere on his upper body. This is explained in Vana Parva of Mahabharata when Duryodhana wants to starve himself to death following his shameful defeat at the hands of Chitrasena in front of the Pandavas. The danavas realizing this summon him to patala and explain him his true purpose.
"The Danavas said, 'O Suyodhana, O great king? O perpetuator of the race of Bharata, thou art ever surrounded by heroes and illustrious men. Why hast thou, then, undertaken to do such a rash act as the vow of starvation? The suicide ever sinketh into hell and becometh the subject of calumnious speech. Nor do intelligent persons like thee ever set their hands to acts that are sinful and opposed to their best interests and striking at the very root of their purposes. Restrain this resolve of thine, therefore, O king, which is destructive of morality, profit, and happiness, of fame, prowess, and energy, and which enhanceth the joy of foes O exalted king, know the truth, the celestial origin of thy soul, and the maker of thy body, and then summon thou patience to thy aid. In days of old. O king, we have obtained thee, by ascetic austerities from Maheswara. The upper part of thy body is wholly made of an assemblage of Vajras, and is, therefore, invulnerable to weapons of every description, O sinless one. The lower part of thy body, capable of captivating the female heart by its comeliness was made of flowers by the goddess herself--the wife of Mahadeva. Thy body is thus, O best of kings, the creation of Maheswara himself and his goddess. Therefore, O tiger among kings, thou art of celestial origin, not human.
So while Duryodhana can be defeated in battle he cannot be killed, which is why Bhima had to strike him below the waist following Krishna's advice:
"'Vasudeva said, "The instruction received by them hath been equal. Bhima, however, is possessed of greater might, while the son of Dhritarashtra is possessed of greater skill and hath laboured more. If he were to fight fairly, Bhimasena will never succeed in winning the victory. If, however, he fights unfairly he will be surely able to slay Duryodhana. The Asuras were vanquished by the gods with the aid of deception. We have heard this. Virochana was vanquished by Shakra with the aid of deception. The slayer of Vala deprived Vritra of his energy by an act of deception. Therefore, let Bhimasena put forth his prowess, aided by deception! At the time of the gambling, O Dhananjaya, Bhima vowed to break the thighs of Suyodhana with his mace in battle. Let this crusher of foes, therefore, accomplish that vow of his. Let him with deception, slay the Kuru king who is full of deception. If Bhima, depending upon his might alone, were to fight fairly, king Yudhishthira will have to incur great danger.
You can further read this answer to understand how this act of Bhima is justified.