### Brain Teasers

# Class Photo

Jacob's class picture has 40 photos arranged in a 8 x 5 grid. The photos in the top row are numbered 1 through 8 from left to right, with the photos in the remaining rows similarly numbered (as shown below). Given the following clues (bordering includes horizontal, vertical, and diagonal), where is Jacob's picture?

X X X X X X X X (1 - 8)

X X X X X X X X (9 - 16)

X X X X X X X X (17 - 24)

X X X X X X X X (25 - 32)

X X X X X X X X (33 - 40)

1) There are 20 boys and 20 girls.

2) Each row and column has at least two girls, but no more than four girls.

3) Every girl borders at least one other girl.

4) Girls are located at positions that are prime numbers.

5) Boys are located at positions that are either squares or cubes.

6) Jacob is the only boy that borders a unique number of girls.

X X X X X X X X (1 - 8)

X X X X X X X X (9 - 16)

X X X X X X X X (17 - 24)

X X X X X X X X (25 - 32)

X X X X X X X X (33 - 40)

1) There are 20 boys and 20 girls.

2) Each row and column has at least two girls, but no more than four girls.

3) Every girl borders at least one other girl.

4) Girls are located at positions that are prime numbers.

5) Boys are located at positions that are either squares or cubes.

6) Jacob is the only boy that borders a unique number of girls.

### Hint

The number 1 is not a prime number.### Answer

Per clue 4: girls are at positions 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and 37. Per clue 5: boys are at positions 1, 4, 9, 16, 25, 36, 8, and 27.B G G B G _ G B

B _ G _ G _ _ B

G _ G _ _ _ G _

B _ B _ G _ G _

_ _ _ B G _ _ _

Per clue 2: make sure each row/column has at least two, but no more than four, girls. This puts boys at positions 6 and 21, and a girl at position 33.

B G G B G B G B

B _ G _ G _ _ B

G _ G _ B _ G _

B _ B _ G _ G _

G _ _ B G _ _ _

There are still seven girls to identify, which must be assigned to columns 2, 4, 6, and 8 (clues 1 and 2). This means that boys must fill the open spaces in the other columns (positions 15, 35, and 39).

B G G B G B G B

B _ G _ G _ B B

G _ G _ B _ G _

B _ B _ G _ G _

G _ B B G _ B _

Since all girls must border at least one other girl (clue 3) and only one more girl can be added to column 2, the next placements can be made (girls at positions 14 and 26).

B G G B G B G B

B _ G _ G G B B

G _ G _ B _ G _

B G B _ G _ G _

G _ B B G _ B _

The rest of column 2 must be boys (positions 10, 18, and 34).

Because no row can have more than 4 girls, each row must have exactly 4 girls. Girls must then be at positions 12, 38, and 40. Boys must then be at positions 22 and 30.

B G G B G B G B

B B G G G G B B

G B G _ B B G _

B G B _ G B G _

G B B B G G B G

The final two girls must be at either positions 20/32 or 24/28 (clue 2). If the first combination is chosen, then there will be no boy that borders a unique number of girls (clue 6). By placing the girls at the second combination (and boys at the first combination), clue 6 will be satisfied.

B G G B G B G B

B B G G G G B B

G B G B B B G G

B G B G G B G B

G B B B G G B G

The boys in the first row border (1, 5, 4, 1) girls.

The boys in the second row border (2, 5, 4, 3) girls.

The boys in the third row border (4, 6, 5, 5) girls.

The boys in the fourth row border (3, 3, 5, 4) girls.

The boys in the fifth row border (2, 2, 3, 3) girls.

The only boy that borders a unique number of girls is the boy who borders six girls, which is at position 20. Therefore, Jacob is at position 20.

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