Knightmare

At a glance

The possible move numbers and their frequencies are:

Thus the complete set of possible move numbers is 6, 8, 10, 12, 13, 16, 20, and 24. The most common move number is 16, appearing in 144 rooms.

Key idea or plan

Do not classify rooms by all 512 exact addresses. Instead, classify each coordinate by how the cube’s two walls affect a change of 1 and a change of 2. Only three coordinate behaviors occur. Combining three such behaviors produces ten room profiles, and those same profiles control both the legal moves and the number of rooms.

1. Find every possible move number

For one coordinate, only its distance from the nearer wall matters. There are three useful coordinate types:

• E (edge): 1 or 8. There is one legal direction for a change of 1 and one legal direction for a change of 2.
• N (near): 2 or 7. There are two legal directions for a change of 1 but only one legal direction for a change of 2.
• I (interior): 3, 4, 5, or 6. There are two legal directions for either change.

Let a coordinate’s pair be (one-step choices, two-step choices). Then E has (1,1), N has (2,1), and I has (2,2).

For each ordered choice of two different coordinate axes, multiply the number of legal two-step directions on the first axis by the number of legal one-step directions on the second. Adding over all six ordered axis pairs gives the room’s move number.

The order of the three coordinate types does not affect the result, so every room belongs to one of ten profiles:

Reading the distinct totals from the table gives exactly the eight possible move numbers stated above.

2. Count the rooms of each kind

Along any axis there are 2 E coordinates, 2 N coordinates, and 4 I coordinates. We also choose which axes receive the listed types.

Combining profiles with the same move number gives 72 rooms with move number 10 and 144 rooms with move number 16. The other rows already have distinct move numbers. Therefore 16 is the most common.

3. Prove the classification is complete

Every coordinate from 1 through 8 is in exactly one of E, N, or I. Therefore every room address has exactly one unordered three-letter profile from the ten listed profiles, so every room is covered once.

For a move, the destination’s coordinate differences uniquely identify the axis changed by 2, the different axis changed by 1, both signs, and the unchanged axis. Thus the ordered-axis count cannot count one destination twice.

Finally, the profile totals add to

8 + 24 + 48 + 24 + 96 + 96 + 8 + 48 + 96 + 64 = 512.

That is the total number of rooms in the cube, confirming that the classification is exhaustive.

Check

An interior room such as (3, 3, 3) has 6 ordered choices for the axis changed by 2 and the different axis changed by 1. Each change has 2 sign choices, so it has 6 × 2 × 2 = 24 destinations, matching the III row.

A corner such as (1, 1, 1) still has 6 ordered axis choices, but every nonzero change has only one legal sign, so it has 6 destinations, matching the EEE row.

The final frequency total is 8 + 24 + 72 + 8 + 96 + 144 + 96 + 64 = 512, exactly the number of cube rooms. The smallest, largest, and total counts therefore agree with independent features of the original setup.

Teaching note

A complete student solution need not use the letters E, N, and I or the displayed formula. Any organization is acceptable if it distinguishes the same three boundary behaviors, accounts for all ten coordinate-type profiles, produces the correct aggregated frequency table, and proves the three requested completeness conditions.

Common wrong turns

• Counting a move that changes all three coordinates. The changes must be 2, 1, and 0 in some order.
• Treating a coordinate value of 2 or 7 like a fully interior value. It allows both one-step directions but only one two-step direction.
• Forgetting that EEN, ENE, and NEE are different placements of the same profile when counting rooms.
• Listing the ten profile totals without combining profiles that share move number 10 or 16.
• Showing that the listed cases work but not proving that every coordinate belongs to one of the three types.