### Brain Teasers

# Class Photo 2

Kevin's class picture has 28 photos arranged in a 7 x 4 grid. The photos in the top row are numbered 1 through 7 from left to right, with the photos in the remaining rows similarly numbered. Given the following clues (bordering includes horizontal, vertical, and diagonal), where is Kevin's picture?

X X X X X X X (1 - 7)

X X X X X X X (8 - 14)

X X X X X X X (15 - 21)

X X X X X X X (22 - 28)

1) There are 11 boys and 17 girls.

2) No boy is next to another boy horizontally or vertically.

3) There are no more than three boys in each row.

4) One column has only girls.

5) Three boys are not along the edge of the picture.

6) The second boy is at an odd position.

7) The range of the boys' positions is 24.

8) The median of the boys' positions is 12 or 13 or 14.

9) There are boys in two corners.

10) Kevin occupies an even position, and is the only boy who borders a unique number of girls.

X X X X X X X (1 - 7)

X X X X X X X (8 - 14)

X X X X X X X (15 - 21)

X X X X X X X (22 - 28)

1) There are 11 boys and 17 girls.

2) No boy is next to another boy horizontally or vertically.

3) There are no more than three boys in each row.

4) One column has only girls.

5) Three boys are not along the edge of the picture.

6) The second boy is at an odd position.

7) The range of the boys' positions is 24.

8) The median of the boys' positions is 12 or 13 or 14.

9) There are boys in two corners.

10) Kevin occupies an even position, and is the only boy who borders a unique number of girls.

### Answer

A) There are five boys prior to the median, meaning that there are three boys in each of the first two rows. If the first boy is at location 4, then there could only be two boys in the first row. The first boy must be located before position 4, therefore a girl must occupy: 28 (clue 7). The second boy can not be at position 4 (clue 6). The earliest position that the third boy could occupy is position 5. Therefore, a girl must occupy: 4.B) If boys are at positions 1 (a corner) and 25 (clue 7), then a boy would have to be at either position 7 or 22 (clue 9). In either case, more than three boys would not be along the edge of the picture, violating clue 5. Therefore, a girl must occupy: 1. Boys must occupy: 7 and 22 (clue 9). Girls must occupy: 6, 14, 15, and 23.

C) For three boys to be in the first row, the second boy must occupy: 5 (which also satisfies clue 6). A girl must occupy 12. Per clue 8, a boy must occupy: 13. A girl must occupy: 20. To avoid violating clue 5, boys must occupy: 8 and 21. A girl must occupy: 9.

D) Clues 8 and 9 have now been satisfied, and 16 of the 28 positions have been filled. To satisfy clues 1 through 7, the last 12 positions can be filled in four different ways. However, only one of those ways will satisfy clue 10. In the three ways that do not work, the boy that borders a unique number of girls is located at an odd position.

Step A Step B Step C Step D

xxxGxxx GxxGxGB GxxGBGB GBGGBGB

xxxxxxx xxxxxxG BGxxGBG BGBGGBG

xxxxxxx Gxxxxxx GxxxxGB GBGGGGB

xxxxxxG BGxxxxG BGxxxxG BGBGBGG

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

The boys in the second row border (3, 6, 5) girls.

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

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

The only boy that borders a unique number of girls is the boy who borders six girls, which is at position 10. Kevin is at position 10.

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