solution: The first remainder is even with probability $2/6=1/3$ and odd with probability 2/3.  The second remainder is even with probability $3/6=1/2$ and odd with probability 1/2.  The parity of the first remainder and the parity of the second remainder are independent, since they're determined by separate spins of the wheel.
  
  The shaded squares are those that indicate that both remainders are odd or both are even. Hence the square is shaded with probability \[
  \frac{1}{3}\cdot \frac{1}{2} + \frac{2}{3}\cdot\frac{1}{2} =\boxed{\frac{1}{2}}.
  \]
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  The wheel shown is spun twice, so that the numbers indicated by the pointer are randomly determined (with each number on the wheel being equally likely). The two numbers determined in this way are recorded. The first number is divided by 4, determining one of the remainders 1,2,3 marking the columns of the checkerboard shown. The second number is divided by 5, determining one of the remainders 1,2,3,4 marking the rows of the checkerboard. Finally, a checker is placed on the square where this column and row meet. What is the probability that the checker is placed on a shaded square of the checkerboard?
  
  [asy]
  unitsize(1cm);
  draw(Circle((0,0),2),linewidth(0.7));
  draw((1.7,1)--(-1.7,-1),linewidth(0.7));
  draw((1.7,-1)--(-1.7,1),linewidth(0.7));
  draw((0,2)--(0,-2));
  label("1",(0.8,0.5),NW);
  label("2",(0.8,-0.5),SW);
  label("6",(-0.8,0.5),NE);
  label("9",(-0.8,-0.5),SE);
  label("3",(-0.7,0),W);
  label("7",(0.7,0),E);
  draw((-2.8,0)--(-2.1,0),Arrow);
  label("Pointer",(-2.8,0),W);
  fill((3,0)--(3,1)--(4,1)--(4,0)--cycle,gray(0.7));
  fill((3,-2)--(3,-1)--(4,-1)--(4,-2)--cycle,gray(0.7));
  fill((4,1)--(4,2)--(5,2)--(5,1)--cycle,gray(0.7));
  fill((4,-1)--(4,0)--(5,0)--(5,-1)--cycle,gray(0.7));
  fill((5,0)--(5,1)--(6,1)--(6,0)--cycle,gray(0.7));
  fill((5,-2)--(5,-1)--(6,-1)--(6,-2)--cycle,gray(0.7));
  draw((3,-2)--(3,2)--(6,2)--(6,-2)--cycle,linewidth(0.7));
  draw((3,-1)--(6,-1),linewidth(0.7));
  draw((3,0)--(6,0),linewidth(0.7));
  draw((3,1)--(6,1),linewidth(0.7));
  draw((4,-2)--(4,2),linewidth(0.7));
  draw((5,-2)--(5,2),linewidth(0.7));
  label("1",(3.5,-2),S);
  label("2",(4.5,-2),S);
  label("3",(5.5,-2),S);
  label("1",(3,-1.5),W);
  label("2",(3,-0.5),W);
  label("3",(3,0.5),W);
  label("4",(3,1.5),W);
  [/asy]
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: \frac{1}{2}
extra_info.question: The wheel shown is spun twice, so that the numbers indicated by the pointer are randomly determined (with each number on the wheel being equally likely). The two numbers determined in this way are recorded. The first number is divided by 4, determining one of the remainders 1,2,3 marking the columns of the checkerboard shown. The second number is divided by 5, determining one of the remainders 1,2,3,4 marking the rows of the checkerboard. Finally, a checker is placed on the square where this column and row meet. What is the probability that the checker is placed on a shaded square of the checkerboard?
  
  [asy]
  unitsize(1cm);
  draw(Circle((0,0),2),linewidth(0.7));
  draw((1.7,1)--(-1.7,-1),linewidth(0.7));
  draw((1.7,-1)--(-1.7,1),linewidth(0.7));
  draw((0,2)--(0,-2));
  label("1",(0.8,0.5),NW);
  label("2",(0.8,-0.5),SW);
  label("6",(-0.8,0.5),NE);
  label("9",(-0.8,-0.5),SE);
  label("3",(-0.7,0),W);
  label("7",(0.7,0),E);
  draw((-2.8,0)--(-2.1,0),Arrow);
  label("Pointer",(-2.8,0),W);
  fill((3,0)--(3,1)--(4,1)--(4,0)--cycle,gray(0.7));
  fill((3,-2)--(3,-1)--(4,-1)--(4,-2)--cycle,gray(0.7));
  fill((4,1)--(4,2)--(5,2)--(5,1)--cycle,gray(0.7));
  fill((4,-1)--(4,0)--(5,0)--(5,-1)--cycle,gray(0.7));
  fill((5,0)--(5,1)--(6,1)--(6,0)--cycle,gray(0.7));
  fill((5,-2)--(5,-1)--(6,-1)--(6,-2)--cycle,gray(0.7));
  draw((3,-2)--(3,2)--(6,2)--(6,-2)--cycle,linewidth(0.7));
  draw((3,-1)--(6,-1),linewidth(0.7));
  draw((3,0)--(6,0),linewidth(0.7));
  draw((3,1)--(6,1),linewidth(0.7));
  draw((4,-2)--(4,2),linewidth(0.7));
  draw((5,-2)--(5,2),linewidth(0.7));
  label("1",(3.5,-2),S);
  label("2",(4.5,-2),S);
  label("3",(5.5,-2),S);
  label("1",(3,-1.5),W);
  label("2",(3,-0.5),W);
  label("3",(3,0.5),W);
  label("4",(3,1.5),W);
  [/asy]
extra_info.split: test