bigant146 wrote: Given a rectangular table with 2 rows and 100 columns. Dima fills the cells of the first row with numbers 1, 2 or 3. Prove that Alex can fill the cells of the second row with numbers 1, 2, 3 in such a way that the numbers in each column are different and the sum of the numbers in the second row equals 200. $\color{blue}\boxed{\textbf{Proof:}}$ $\color{blue}\rule{24cm}{0.3pt}$ Let $x$ be the number of $1's$, $y$ the number of $3's$ and $z$ the number of $2's$ $\Rightarrow x+y+z=100...(I)$ Let $S$ be the sum of the numbers in the second row Let $2_i$ the $i$-th $2's$ Let "$a\to b$" the transformation that Alex makes, that is, if Dima places $a$ above, Alex places $b$ below $\color{red}\boxed{\textbf{If the number of 2's is even}}$ $\color{red}\rule{24cm}{0.3pt}$ Numbers of $2's=z=2k$ Alex's strategy is: $$2_{2i-1}\to 1, \forall 1\le i\le k$$$$2_{2i}\to 3, \forall 1\le i\le k$$$$1\to 2$$$$3\to 2$$$$\Rightarrow S=(1)(k)+(3)(k)+(2)(x)+(2)(y)$$$$\Rightarrow S=4k+2x+2y$$$$\Rightarrow S=2x+2y+2z$$By $(I):$ $$\Rightarrow S=200_\blacksquare$$$\color{red}\rule{24cm}{0.3pt}$ $\color{red}\boxed{\textbf{If the number of 2's is odd}}$ $\color{red}\rule{24cm}{0.3pt}$ Numbers of $2's=z=2k+1$ If there is at least one $1$ Alex's strategy is: $$2_{2i-1}\to 1, \forall 1\le i\le k$$$$2_{2i}\to 3, \forall 1\le i\le k$$$$2_{2k+1}\to 1$$$$1\to 2, \text{ except the last one that will become 3}$$$$3\to 2$$$$\Rightarrow S=(1)(k)+(3)(k)+1+(2)(x-1)+3+(2)(y)$$$$\Rightarrow S=4k+2+2x+2y$$$$\Rightarrow S=2x+2y+2z$$By $(I):$ $$\Rightarrow S=200_\blacksquare$$If there is at least one $3$ Alex's strategy is: $$2_{2i-1}\to 1, \forall 1\le i\le k$$$$2_{2i}\to 3, \forall 1\le i\le k$$$$2_{2k+1}\to 3$$$$1\to 2$$$$3\to 2, \text{ except the last one that will become 1}$$$$\Rightarrow S=(1)(k)+(3)(k)+3+(2)(x)+1+(2)(y-1)$$$$\Rightarrow S=4k+2+2x+2y$$$$\Rightarrow S=2x+2y+2z$$By $(I):$ $$\Rightarrow S=200_\blacksquare$$$\color{red}\rule{24cm}{0.3pt}$ $$\Rightarrow \boxed{\textbf{Alex can achieve his goal}}_\blacksquare$$$\color{blue}\rule{24cm}{0.3pt}$