# Prince Rupert's Cube

## Prince Rupert's Cube

Let $C$ be a unit cube.

The largest square tunnel that can be made in $C$ through which a larger cube may be passed has sides of length:

- $\dfrac {3 \sqrt 2} 4 = 1 \cdotp 06066 \, 0$

This sequence is A093577 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Proof

## Source of Name

This entry was named for Prince Rupert of the Rhine.

## Historical Note

According to John Wallis, the puzzle now known as Prince Rupert's Cube was first posed by Prince Rupert of the Rhine in $1693$.

The correct answer was determined by Pieter Nieuwland.

An incorrect solution to this puzzle, often quoted in the literature, was provided by Wallis himself, who assumed that the tunnel in question would be parallel to the space diagonal of the cube.

This provides a solution of $\sqrt 6 - \sqrt 2 \approx 1 \cdotp 03527$.

## Sources

- 1950: D.J.E. Schrek:
*Prince Rupert's problem and its Extension by Pieter Nieuwland*(*Scripta Math.***Vol. 16**: 73 – 80)

- 1950: D.J.E. Schrek:
*Prince Rupert's problem and its Extension by Pieter Nieuwland*(*Scripta Math.***Vol. 16**: 261 – 267)

- 1961: H.M. Cundy and H.M. Cundy:
*Mathematical Models*(2nd ed.)

- 1983: François Le Lionnais and Jean Brette:
*Les Nombres Remarquables*... (previous) ... (next): $1,06066 0172 \ldots$

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $1 \cdotp 060 \, 660 \ldots$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1 \cdotp 06066 \, 0 \ldots$