(1) drawings
Draw a circle, assuming its area is 2 n, paint half of its mask black and the remaining area is 2 (n- 1), then paint the remaining area black as 1/2, and repeat the above process until the last remaining area is 2, then paint half black and the remaining area is 1.
Now let the blackened area be equal to the number of rice grains, and the blackened area of each operation corresponds to the number of rice grains in each cell backwards. Then, the number of rice grains in the 64th unit corresponds to the first operation. Through the solution, it can be seen that there are 2 63 grains in the 64th grid, so 2 (n-1) = 2 63, and n = 64.
Total black area = circular area-final area without black coating = 2 n- 1 = 2 64- 1.
Please draw it yourself and think for yourself.
(2) dislocation subtraction summation
Suppose the total number of rice grains is S.
S = 1+2+...+2 63 (there are 2 (n- 1) grains in the nth grid).
2S= 2+2^2+...+2^63+2^64
2S-S=(2+2^2+...+2^63+2^64)-( 1+2+......+2^63)=2^64- 1=S
By using dislocation subtraction, the summation formula of equal ratio sequence can be derived. The principle is similar to this problem. Say it simply.