and-thirty and a quarter that I paid on your own behalf, together with half of what remains of what I paid under my guarantee, after deducting the twenty-three and three eighths and a sixteenth that you paid under yours. That half is three and seven eighths and a thirty-second, bringing the total to thirty-five and an eighth and a thirty-second,” as we have stated. Once he has recovered this amount from him, he is left bearing twenty-seven and two eighths and three thirty-seconds of what he paid under his guarantee. He recovers this from the sixth man when he encounters him, thereby recovering the full amount he paid under his guarantee on behalf of them all.

Section

If the fifth man, from whom the first recovered eighteen and three quarters, the second thirty-four and three eighths, the third thirty-nine and a sixteenth, and the fourth thirty-five and an eighth and a thirty-second, then encounters the sixth man, from whom the first recovered six and an eighth, the second fifteen and five eighths, the third twenty-three and three eighths and a sixteenth, and the fourth twenty-seven and two eighths and three thirty-seconds, he recovers twenty-seven and two eighths and three thirty-seconds from him. He says to him: “I have borne one hundred and twenty-seven and two eighths and three thirty-seconds in payments to the first, second, third, and fourth men. Of this, one hundred was due from me, and I cannot recover it from anyone. The remaining twenty-seven and two eighths and three thirty-seconds I paid under my guarantee on your behalf, so pay it to me.” Once he has recovered this amount from him, he has recovered the full amount he paid under his guarantee on behalf of them all.

When this amount has been recovered from the sixth man, he will have paid a full hundred, as was due from him under the original debt and as each of the others has paid. For he paid six and a quarter to the first, fifteen and five eighths to the second, twenty-three and three eighths and a sixteenth to the third, twenty-seven and two eighths and three thirty-seconds to the fourth, and twenty-seven and two eighths and three thirty-seconds to the fifth. All this amounts to one hundred, as we have stated.

Thus, praise be to Allah, we have fulfilled our undertaking to explain the problem in the two ways mentioned. Whoever understands this and grasps its meaning will have no difficulty knowing how to proceed, whatever the order in which they encounter one another. The possible orders are numerous, difficult to count, and lengthy to examine exhaustively. Most of the scholars—may Allah have mercy on them—would not read it