To solve the Muhammad puzzle correctly, begin with a young man in Mecca with a very big dream. Leading trade caravans to Damascus as well as his interaction with caravans crisscrossing Mecca opened his eyes to a world much larger than his own. He knew that his family, beginning with his ancestor Qusay and great-grandfather Hashim, had successfully led their Quraysh tribe but that leadership had deteriorated in the generations preceeding his birth. Other Arab leaders had successfully united their tribes and formed kingdoms - there were at least eight at the time throughout Arabia and Yemen - in a way that no-one in Mecca had been able to do. Arab armies had for the first time gained an impressive vicory over the mighty Perisan Empire in present-day Iraq, and forced Ethiopian invaders from Yemen. Why could he not also unite the tribes of Mecca under his leadership and form a great kingdom?
It was a dream that developed into a powerful concept as the young Muhammad approached the age of forty. From Christians and Jews he learned that the great King David had been accepted by his people as a Prophet before he was crowned King at the same age. From them he also learned that one invisible God who spoke through his Prophet was much more powerful than the flimsy tribal gods whose images filled the Kaaba. He, Muhammad, would also become that Prophet-King, and his tribe would accept him as their leader with the words of Allah flowing through him to confirm his authority.
His initial message was a simple one, "I am the Prophet of Allah, who wants you to follow me. If you don't, he will be very angry at you." His promise to the many slaves of Mecca that obeying him would end in their enjoying the treasures of the Roman and Persian Empires (Bukhari vol 8 book 78 nrs 625, 626) gained him some converts. Their loyalty to their slaveowners was now divided, with the Quraysh soon turning against Muhammad instead of accepting him as their Prophet and King as he had anticipated. His messages grew more and more stident, with the short Quranic Meccan suras containing more than 140 verses vividly describing the hell awaiting those who refused him.
When it became clear that his own tribe would not accept him, he approached others for the same purpose. The Thaqif in Ta'if were no more interested than the Quraysh, and neither was any other tribe. After 11 difficult years, Muhammad met a few men from the Aus and Khazraj tribes in Medina, led by Abbas Ibn Ubada, who were envious of the prosperity of the Jewish tribes who had lived there for centuries. The Aus and Khazraj often robbed the Jews, who retaliated by warning that a Prophet would come to punish them. Abbas and his comrades responded to Muhammad's message by concluding they should attract him to their side before the Jews did. They pledged their allegiance and invited him to Medina as their leader. Muhammad gladly accepted, naively convinced the Jews there would also accept him as a Prophet.
Among those who migrated to Medina with Muhammad from Mecca was a group of common highwaymen known as the Saalik (link to "The Rise of Islam" and "The Earlier Followers of Mohammed at Mecca"). They had only been in Medina a few months before Muhammad began sending them as well as other Muslims on raids against caravans passing through the region. The Quraysh responded to these raids by sending out their armies against the Muslim forces, but were soundly defeated. Emboldened by his victories, Muhammad turned against the Jews of Medina and quickly exiled and murdered all of them. He continued to claim that Allah was speaking directly through him with revelations that allowed him to do whatever he wanted, from raping the wives of his victims to declaring unending war against anyone who opposed him. His promise of a sensual paradise to all who died in his wars was a powerful motivation to his warriors, and by the time Muhammad died at the age of 62 his dream of becoming ruler of Arabia had become true.
Now imagine putting together the jigsaw puzzle (perhaps even one portraying the face of Muhammad!) from the perspective of a Muslim believer or a western non-Muslim apologist for Islam (present and all recent American Presidents included). You believe that a noble young man of Arabia, heartbroken over the idolatry and human rights abuses of his people, was called by God to be the final Prophet of the one true religion. This peaceful Prophet suffered oppression and persecution for 13 years until he was driven from his home to a distant city. There he received direction from God to lead battles of defense against the persecutors who were determined to destroy him and his nascent community. He was guided to conduct raids against the trade caravans to recover the goods his followers had lost in Mecca. He was commanded to remove the Jewish tribes of Medina who opposed his righteous message. He was ordered to marry numerous wives to fortify his political and social unions, as well as to exemplify mercy to widows and divorcees. He was led to order the extermination of all Christians and Jews from Arabia, and was given God's plan to bring the entire world under submission to Islam.
It's not long, however, before you are forcing pieces of the puzzle to try to make them fit. How does a holy Prophet of God in his mid-50s sexually conquer and ejaculate into an innocent 9 year old child? Push that piece into place if you can. How does he tire of her within a few months and begin taking other wives? Force that piece into the puzzle (the first was the beautiful Hafsah, at 20 twice Aysha's age) . Imagine young men on a camel caravan returning from a successful trip to Damascus where they traded olives and figs for the goods their families would live on for the next year. Suddenly they are terrified by a calvary of men with swords held high shouting Allahu Akbar as they swoop down on them to steal their goods, kill some of them, and sell the others for slaves or ransom. Justify that as you push that piece into place.
Listen to the Prophet describe the beautiful virgins waiting for his warriors in paradise to satisfy all their sexual desires if they die in battle, (Quran 52:20 and numerous similar verses) and then sending them off to war to do just that. Push that piece into place. Watch him sending his female prisoners of war across the desert to be sold as slaves for weapons and horses. How do you force that piece into the puzzle? See him and his warriors lining up the young Jewish boys of Medina and examining them for pubic hair to decide whether they would be beheaded with their fathers or exiled with their mothers. Can you push that piece into place (Book 38, 4390)? Listen to the Prophet joyfully telling one wife that Allah has commanded him to marry his own daughter-in-law. Push that into place if you can. Go with him to visit another wife and then hear him lie to get her out of the house so he can sleep with the beautiful Egyptian slave girl he has recently given her. Can you force that into place? Watch him torture a young Jewish man to death for refusing to reveal the location of his wealth that Muhammad wants to steal, and then accompany the Prophet as he "marries" his wife the same night. Can you really push that piece into place? Listen to the Prophet bless and congratulate the companion who drove his spear through the body of a young Jewish poetess as she lay sleeping with her nursing child because she had written poetry he did not like. Try to force that piece into place if you can.
Some jigsaw puzzles have 1000 pieces, and one could easily find that many examples of Muslims trying to force each one into place as they attempt to defend their Prophet. There is, however, good news. Thousands of men and women who were born Muslim are giving up the impossible task (see here and here). They might be now Buddhist, Christian, atheist, or simply following their own conscience, but they are free. They come from many backgounds, but they could probably all echo the sentiments of Aayan Hirsi Ali who was once asked by Manji Irshad why she had chosen to leave Islam rather than become a "Muslim reformer". Aayan's simple response was, "I couldn't live with the dichotomy for five minutes." Many others are saying the same.