Duell with Fate

According to legend, several centuries ago, a distant ancestor of yours committed a daring act by stealing a mystical tarot deck from Fate herself. As a result, your family became renowned worldwide as the greatest fortune tellers. However, this fortune came at a tremendous price. Every 23 years, a mysterious door materializes within your home, and one member of your family must confront Fate in a duel governed by arcane rules known only to your opponent.

Throughout the generations, the outcome has remained unchanged: Fate claims the soul of your family member. Twenty-three years ago, your father accepted the challenge and tragically lost. Now, it is your turn. As you anxiously pace in the dueling room, you stumble upon a revelation that may alter everything: your father had concealed a camera in the room. The tape reveals his arrangement of the 22 cards from the major Arcana.

Fate then introduces an additional card, the Soul, and speaks:

"You may select any card present on the table that has at least one factor. I will then claim every factor on the chosen card. Take the Tower, for instance. It holds a value of 16, with factors of 1, 2, 4, and 8. Those factors now belong to me, resulting in a score of 16 to 15 in my favor. If your total surpasses mine at the end of the game, you shall emerge victorious. However, if you exhaust all your moves, I will claim the remaining cards."

As the game progresses, Fate mentions that if your father were to possess the 10, he would receive great fortune; the 6 would grant true love, and the 2 would break the family curse forever. Nevertheless, none of these possibilities matter as his total falls short in the end.

As the static-filled video concludes, Fate materializes before you. In a dreamlike state, you hear her offer you the Tower to commence the game. How can you possibly defeat Fate and secure other favorable outcomes for yourself?

Take a moment to ponder the solution. Answer in 3, 2, 1.

To outmaneuver Fate, you must adopt a long-term perspective. Accepting the 16 grants Fate the 1, 2, 4, and 8. Furthermore, she effectively claims the 23, 19, 17, and 13 since prime numbers cannot be taken once the 1 is off the table. Smaller primes like 11, 7, 5, and 3 will also belong to Fate, but you can acquire their multiples, such as 22 and 21. Hence, these are the remaining cards in play. Since Fate receives at least one card for each one you take, you can acquire a maximum of half of the remaining cards, rounded down.

Consequently, the best achievable outcome is obtaining the seven largest cards. When combined with the 16, your total would amount to 138. Interestingly, Fate's total would also be 138. In case of a tie, Fate emerges victorious. It becomes apparent that by offering your ancestors the 16 before explaining the complete rules, Fate manipulated the game to guarantee her triumph. Nevertheless, thanks to your father's foresight, you are now aware of the truth.

Speaking of your family, if you desire to claim the 2 and break the curse, you must do so on your first turn since it is a prime number. If you succeed, Fate will acquire the 1, and once again, she will secure the higher primes. In theory, you can obtain at most 8 of the remaining 17 cards. Selecting the eight largest would grant you a nerve-wrackingly narrow victory of 140 to 136. But is it attainable?

To minimize the number of cards Fate receives for each one you take, you can start with the 22 and 14. With the 7 removed, you can then acquire the 21, followed by the 15. You still require these cards, and claiming the next one will grant Fate two cards. However, this is not an issue. For instance, if you choose the 16, Fate will receive the 4 and 8, but you will still gain the 12, 18, and 20. By employing similar reasoning, you could claim the 23 on your initial turn and secure the 10 or the 6.

Nevertheless, claiming more than one reward is impossible. Therefore, must you choose between wealth, love, or breaking the curse? Fate awaits, but the decision ultimately rests with you.