An aged merchant of Baghdad was much respected by all who knew him. He had three sons, and it was a rule of his life to treat them equally. Whenever one son received a present, the other two each received a present of equal value. One day this worthy man fell sick and died, bequeathing all of his possessions to his three sons in equal shares. The only difficulty that arose was over the stock of honey. There were exactly 33 barrels. The old man left instructions that each son should not only receive an equal quantity of honey, but each son should receive exactly the same number of barrels, and no honey should be transferred from barrel to barrel on account of the waste involved. Now, as 10 of these barrels were full of honey, 13 were half full, and 10 were empty, this was quite a puzzle, especially because each brother objected to taking less than 2 or more than 7 barrels of the same description (full, half full, or empty). Solve this puzzle with an integer mode

Son number 1 has 3 full, 5 half, and 3 empty barrels.

Son number 2 has 4 full, 3 half and 4 empty barrels.

Son number 3 has 3 full, 5 half, and 3 empty barrels.

Step-by-step explanation:

Let's first calculate the amount of honey we have (in barrel unit):

10*1 + 13*0.5 + 10*0 = 16.5 honey units

As the father wants to split it equally between the 3 sons, each of them should get 16.5 / 3 = 5.5 honey units

And since we have 33 barrels in total which must be split equally between the sons, each should receive 33 / 3 = 11 barrels.

Since the sons are identical, let's start by splitting each of the barrels equally. We can rearrange later to match all criteria:

3 full, 4 half, and 3 empty for each of the sons, makes it 10 barrels and 3*1 + 4*0.5 + 3*0 = 5 honey unit

For 3 sons there would be 1 full, 1 half and 1 empty barrels left

We can give 1 half barrel to the 1st son, now he has 11 barrels and 5.5 units of honey. He's good to go. we have 1 full, 1 and 1 empty barrels left.

Give the full barrel to the 2nd son, now he has 11 barrels, but 6 unit of honey. So we trade the empty barrel for a half barrel of his. So he would have 4 full, 3 half and 4 empty barrels. That accounts for 11 barrels, 4*1 + 3*0.5 + 3*0 = 5.5 barrels.

We are left with 1 half barrel, that can be given to the 3rd son, now he would have 3 full, 5 half, and 3 empty barrels. That accounts for 11 barrels and 3*1 + 5*0.5 + 3*0 = 5.5 unit of honey.