, 2026

We can systematically count the number of squares by considering
the possible sizes of the squares.

Case 1: Squares of size $1\times1$. There are 34 such squares. Case 2: Squares of size $2\times2$. Begin with a $2\times2$ square in the top left corner of the diagram. Shifting this $2\times2$ square one column to the right gives a second $2\times2$ square. Shifting this second $2\times2$ square one column to the right again gives a third $2\times2$ square. Thus, there are $3$ such $2\times2$ squares within the top two rows of the diagram. Each of these $3$ squares can be shifted down one row to give $3$ more $2\times2$ squares, each contained within the second and third rows of the diagram. Continuing in this way, there are $3$ such $2\times2$ squares within rows three and four, rows four and five, rows five and six, and rows six and seven. Thus, there are $3\times6=18$ such $2\times2$ squares. The first and the last of these $18$ squares are shown in the following diagram. [[IMAGE0]] There are $4$ additional $2\times2$ squares. One of these occupies the leftmost two squares within rows five and six. This square can be shifted down one row, and each of these $2\times2$ squares can be shifted four columns to the right. The first and the last of these 4 squares are shown in the following diagram. [[IMAGE1]] In total, there are $18+4=22$ squares of size $2\times2$. Case 3: Squares of size $3\times3$. Beginning with a $3\times3$ square in the top left corner of the diagram and counting in a similar way, there are $2$ such $3\times3$ squares within the top three rows of the diagram. There are $2$ such $3\times3$ squares within rows two through four, rows three through five, rows four through six, and finally rows five through seven. Thus, there are $2\times5=10$ such $3\times3$ squares. The first and the last of these $10$ squares are shown in the following diagram. [[IMAGE2]] There are $2$ additional $3\times3$ squares. One of these occupies the leftmost three squares within rows five through seven. The second of these occupies the rightmost three squares within rows five through seven. These $2$ squares are shown in the following diagram. [[IMAGE3]] In total, there are $10+2=12$ squares of size $3\times3$. Case 4: Squares of size $4\times4$. There is a $4\times4$ square that occupies all four columns within rows one through four. This square can be shifted downward one row to occupy rows two through five, shifted downward one row again to occupy rows three through six, and finally one more time to occupy rows four through seven. These are the only $4\times4$ squares and thus there are $4$ in total. The first and the last of these $4$ squares are shown in the following diagram. [[IMAGE4]] There are no squares with dimensions greater than $4\times4$, and so there are $34+22+12+4=72$ squares of all sizes in the
diagram.