This Week's Fiddler: Apr 10, 2026

Posted on Apr 10, 2026

Problem

You have four 6-faced number cubes. To display a 3-digit number, you pick three distinct cubes and line them up left to right. Leading zeros are allowed, so for example 7 is displayed as 007.

A 6 can also be turned upside down and used as a 9, so 6 and 9 are effectively the same symbol.

How should the 24 faces be labeled so that you can display every whole number from 1 through \(N\), with no gaps, and how large can \(N\) be?

Main Solution

The key is to count how many cubes each digit must appear on.

Because one-digit numbers are shown with two leading zeros, the digit 0 must appear on at least two cubes.
To make 111, 222, 333, 444, 555, and 666, the symbols 1, 2, 3, 4, 5, and 6/9 must each appear on at least three cubes.
To make 77 and 88, the digits 7 and 8 must each appear on at least two cubes.

If we want to reach at least 776, that already forces at least

\[ 2 + 6\cdot 3 + 2 + 2 = 24 \]

digit-cube incidences. But the four cubes have exactly \(4\cdot 6=24\) faces total, so this uses every face with no slack at all.

Now look one step farther. To make 777, the digit 7 would have to appear on three cubes, not just two. That would require at least 25 incidences overall, which is impossible.

Upper bound: \(N \le 776\).

So the whole problem is whether 776 is actually achievable.

A Working Design

It is. One optimal arrangement is:

Cube A

0 1 2 3 4 5

Cube B

0 1 2 3 4 6/9

Cube C

1 2 5 6/9 7 8

Cube D

3 4 5 6/9 7 8

These four cubes can display every number from 1 through 776. A few examples:

7 = 0 on A, 0 on B, 7 on C
248 = 2 on A, 4 on B, 8 on D
666 = 6/9 on B, 6/9 on C, 6/9 on D
775 = 7 on C, 7 on D, 5 on A

Best possible value: \( \boxed{776} \).

Interactive Checker

Type any number from 1 to 999 and the checker will either show one valid cube assignment or report that the number cannot be made. For this design, the first failure is 777.

Number:

Loading checker...

This Week's Extra Credit

How many distinct unordered cube designs achieve this same optimal value \(N=776\)? Faces on the same cube are not ordered, and swapping two faces on one cube does not create a new design.

Extra Credit Solution

The upper-bound argument does almost all of the structural work.

Symbol	Required cubes
0	2 cubes
1, 2, 3, 4, 5	3 cubes each
6/9	3 cubes
7, 8	2 cubes each

Because these counts already add up to 24 exactly, every optimal design must have exactly this frequency profile. Equivalently, if we describe each cube by the three symbols it is missing, then:

0, 7, and 8 must each be missing from exactly two cubes,
1, 2, 3, 4, 5, and 6/9 must each be missing from exactly one cube.

That shrinks the search space dramatically. After generating all unordered 4-cube designs with exactly that missing-digit pattern, there are only 1075 candidates left to test.

An exhaustive computer check over those 1075 candidates shows that 855 of them can make every number from 1 through 776.

Number of optimal unordered designs: \( \boxed{855} \).

The four cubes displayed above are one of those 855 optimal designs.

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