Here the digital
roots of the six numbers are 6, 4, 1, 2, 7, 9, which together sum to
29, whose digital root is 2.
As the contents of the barrels sold must be a number divisible by 3, if
one buyer purchased twice as much as the other, we must find a barrel
with root 2, 5, or 8 to set on one side.
There is only one barrel, that containing 20 gallons, that fulfils
So the man must have kept these 20 gallons of beer for his own use and
sold one man 33 gallons (the 18-gallon and 15-gallon barrels) and sold
the other man 66 gallons (the 16, 19, and 31 gallon barrels)..