2025 AMC 8 Problem 21

Question

The Konigsberg School has assigned grades

1

through

7

to pods

A

through

G,

one grade per pod. Some of the pods are connected by walkways, as shown in the figure below. The school noticed that each pair of connected pods has been assigned grades differing by

2

or more grade levels. (For example, grades

1

and

2

will not be in pods directly connected by a walkway.) What is the sum of the grade levels assigned to pods

C,

E,

and

F?

12

13

14

15

16

Answer: A

Concepts:graph theory casework logical deduction

Difficulty rating: 1840

Small hint:

The four mutually connected pods must get grades

1,

3,

5,

and

7

Big hint:

Use pod

G

to force where small and large grades can go

Video solution:

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Written solution:

Pods

A,

B,

C,

and

F

are all pairwise connected. Four numbers from

1

through

7

that are all at least

2

apart must be

\{1,3,5,7\}

. Thus

D,

E,

and

G

get the even grades

\{2,4,6\}

.

Pod

G

is connected to

A

and

F

. If

G=4

, then

A

and

F

would have to be

1

and

7

, leaving

B

and

C

as

3

and

5

. But then

E

, which is connected to

C

and

F

, cannot be either remaining even grade without being only

1

away from one of them. So

G

cannot be

4

.

If

G=2

, then

A

and

F

must be

5

and

7

. If

F=5

, then

E

cannot be

4

or

6

, so

F=7

. Also,

C

cannot be

3

, since then

E

again cannot be

4

or

6

. Therefore

C=1

and

E=4

, giving

C+E+F=1+4+7=12

.

The case

G=6

is symmetric, giving

C=7,

E=4,

and

F=1

. The same sum is

12

.

Thus, A is the correct answer.

2025 AMC 8 Problem 21

Answer: A

Difficulty rating: 1840

Small hint:

Big hint:

Video solution:

Written solution:

Problem 21 in other years