A $5\times5$ farm contains houses of six farmers $A$–$F$, as shown. Distribute its nineteen other squares: $D$ receives five, $A,F$ four each, and $B,C,E$ two each. Every allocated square must be in the row or column of its owner\'s house and connected to it directly or through the owner\'s other squares. Fill all blank squares with the owner\'s letter.
Meifeng writes a story in five days, a positive integer number of words each day. Each evening she records the cumulative word count. Dividing these five totals respectively by $1\times2,2\times3,3\times4,4\times5,5\times6$, the fractions sum to five. What is the minimum final word count?
Nine circles are tangent to neighbours as shown, with the centre labeled one. Fill the others with $1,2,3,3,3,4,4,4$, so tangent circles have different labels. How many different fillings are possible?
A shop has souvenirs priced $1,2,\ldots,350$ dollars. Daniela has fifty two-dollar bills and fifty five-dollar bills, with no other money. She buys one souvenir and pays exactly, without change. How many souvenirs can she choose?
Fill a $4\times4$ table using $1$ through $16$ once each, with $16$ fixed top-left. Every row, column and the two main diagonals have the same sum. Find the maximum sum of the six shaded squares.
Two corner squares are removed from a $7\times8$ rectangle in the three displayed ways. For each case, dissect the remaining region into eighteen $1\times3$ or $3\times1$ rectangles, or prove this impossible.
Fill nine boxes using digits $0,1,\ldots,8$ once each in the equation: one-digit number + two-digit number + three-digit number + three-digit number $=999$. Find the minimum positive difference between the two three-digit numbers.
Every ten-digit number using only digits $1,2,3$ is coloured red, green or blue. Any two numbers differing at all ten positions have different colours. If $1111111111$ is red and $1112111111$ blue, what colour is $1231231231$?
