Magnetically Stabilized Puzzle Cube

Introduction

As a competitive speedcuber, I wanted the control of a magnetic cube without making it harder to turn. Magnetic attraction gives each turn a distinct click and helps the layers line up. This project grew from a simple question about the cube’s internal layout: could moving the magnets preserve that feel while improving control?

Can moving the magnets make a puzzle cube easier to turn and help its sides line up?

Cutaway view of a normal puzzle cube showing its central mechanism and surrounding edge and corner pieces. — **Normal cube (side magnets)**Magnets sit along the sides of neighboring edge and corner pieces.

My 3D-printed cube with magnets visible at the roots of the pieces, closer to the center. — **My cube (root magnets)**Magnets sit at the roots of the pieces, closer to the cube’s center.

Process

Internal block geometry and magnetic alignment strategy used to stabilize rotation and reduce failure during high-speed manipulation.

I wanted to preserve the magnetic click while making turns easier to control. Moving magnet mass toward the axis lowers rotational inertia, but it also changes the magnetic pull and its leverage. My original idea was to place the magnets closer to the core and help the layer finish each turn.

I built a Prusa MK3 from a kit, fueled by the included gummy bears, and began printing cubes in my garage. After trying different placements, I put the magnets in the roots of the edges and corners. The later analysis below tests that idea and shows why lower inertia does not automatically mean less overshoot.

Nonlinear Analysis

Think of alignment as the bottom of a bowl: the root-mounted model has a wider, gentler bowl, while the side-mounted model has a narrower, firmer one. The first can guide the layer home from farther away; the second limits its first swing past alignment.

These are model predictions, not prototype measurements. The study treats one turning layer as rigid and compares assumed magnet layouts, not a measured commercial cube; individual-piece wobble, contact, and dry friction are not included.

Selected baseline results
Comparison	Root-mounted	Side-mounted
Nearest ridge around alignment	±20.47°	±11.84°
First overshoot: aligned at 5 rad/s	1.340°	0.778°
Destination from 15° at rest	0°	45°
Small-motion natural frequency	31.99 Hz	57.09 Hz

Root and side magnetic potentials have alignment minima at 0 and 90 degrees, with a shallow unwanted minimum at 45 degrees; circles mark stable minima and crosses mark saddles, with the key below the plot. — Reference potential, Figure 3: both layouts have an unwanted 45° resting position as well as the intended alignments; the ridges alone do not determine capture when the layer starts with speed.

Reduced-order system model

I modeled the positive-z layer—four complete edges, four corners, and one center—as a rigid body about a fixed vertical axis, with angle measured from alignment and the remainder fixed. Inertia includes pieces, magnets, and adhesive mass; magnets follow piece ownership. This uncalibrated 1-DOF model omits dry friction, contact/impact, axial motion, compliance, and independent cubie motion.

Input	Value
Cube envelope	\(56\ \mathrm{mm}\)
Axial cylindrical magnets	48 total: 20 moving and 28 fixed; Ø3 × 2 mm
Aligned face gap	\(0.80\ \mathrm{mm}\)
Remanence \(B_r\)	\(1.20\ \mathrm{T}\)
Viscous damping \(c\)	\(2\times10^{-4}\ \mathrm{N\,m\,s/rad}\)

The state and torque balance are:

\[ \begin{aligned} \mathbf{x}&=(\theta,\omega),\\ \dot{\theta}&=\omega,\\ J\dot{\omega}&=\tau(t)-c\omega-U'(\theta). \end{aligned} \]

Free releases are unforced. Prescribed harmonic excitation uses 30 Hz for periodic-orbit analysis and 10–80 Hz for the frequency sweep:

\[ \begin{aligned} \tau_{\mathrm{free}}&=0,\\ \tau_{\mathrm{forced}}(t)&=0.002\cos(2\pi ft)\ \mathrm{N\,m}. \end{aligned} \]

The finite-cylinder surface-charge model uses prescribed uniform magnetization and sums 560 moving–fixed pair energies; same-assembly interactions are constant. Position and orientation changes produce the nonlinear torque:

\[ \begin{aligned} M&=\frac{B_r}{\mu_0},\\ \tau_{\mathrm{mag}}(\theta)&=-U'(\theta). \end{aligned} \]

Reflection and quarter-turn symmetry extend the 0–45° grid, sampled at 0.1°, into a periodic C² cubic spline. All simulations use this full magnetic potential. Alignment curvature defines the local linearization; a separate even-potential fit over 0–1° gives diagnostic coefficients, with angles in radians:

\[ \begin{aligned} k_0&=U''(0),\\ U'(\theta)&\approx k_{\mathrm{fit}}\theta+\alpha\theta^3+\beta\theta^5. \end{aligned} \]

Parameter	Root-mounted	Side-mounted
\(J\) (kg·m²)	\(1.0916\times10^{-5}\)	\(1.1732\times10^{-5}\)
\(k_0\) (N·m/rad)	\(0.441072\)	\(1.509802\)
\(k_{\mathrm{fit}}\) (N·m/rad)	\(0.441029\)	\(1.509283\)
\(\alpha\) (N·m/rad³)	\(-28.04\)	\(-340.61\)
\(\beta\) (N·m/rad⁵)	\(2167.29\)	\(99597.97\)

Both layouts have positive alignment curvature. For unforced motion:

\[ \begin{aligned} U'(\theta_*)&=0,\\ J\lambda^2+c\lambda+U''(\theta_*)&=0,\\ V&=\tfrac12J\omega^2+U(\theta)-U(\theta_*),\\ \dot V&=-c\omega^2. \end{aligned} \]

The model establishes local asymptotic stability, not global alignment capture. LaSalle’s argument applies within a compact invariant sublevel set containing no other equilibrium; both layouts also admit an attracting 45° equilibrium. The fitted coefficients characterize local behavior but do not replace the full potential in simulation.

Explore the Motion

Choose a starting condition, press Play, or drag the time slider: the two magnet views and every live plot follow the same physical moment.

Interactive playback requires JavaScript; the reference plots are available below.

Capture, Overshoot, and Decay

Local stiffness and inertia explain small-amplitude frequency differences; damped spirals and small-amplitude overshoot also occur in a linear model. Finite barriers and competing capture destinations require the full nonlinear potential.

Capture: the same 15° release finds two different resting positions

Release from 15 degrees at rest: root motion decays toward zero, while side motion approaches 45 degrees. — Report Figure 14: capture from a 15° release at rest.

Released from rest at 15°, the root layout starts inside its 20.47° saddle and returns to 0°; the side layout starts beyond its 11.84° saddle and settles at 45°. Extrapolating the local quadratic potential about 0° would predict alignment capture for both, missing the competing 45° attractor.

Overshoot: a softer pull lets the first swing travel farther

Starting aligned at 5 radians per second, the root model swings farther and more slowly than the side model; both oscillations decay. — Report Figure 6: first overshoot is 1.340° for root and 0.778° for side; this reference shows only the early part of the 1-second release.

Decay: each pass through alignment loses speed

The next positive crossing speed is lower than the current crossing speed for both models; both return curves lie below the no-decay diagonal. — Report Figure 17: the free Poincaré map compares consecutive passes through 0° in the positive direction, not consecutive animation frames.

Near alignment, the next crossing retains about 75.1% of the speed for root and 86.1% for side. Root loses more speed per cycle, but its cycles take longer; this is not the same advantage per second.

Outcome

Figure 11: top cutaway view showing the cube’s internal arrangement. — Assembled puzzle cube demonstrating stabilized internal mechanics and consistent rotational behavior.

Figure 12: perspective view of the assembled puzzle cube. — Assembled puzzle cube demonstrating stabilized internal mechanics and consistent rotational behavior.

The design received U.S. Patent No. 10,583,351 B2 in 2020. The later nonlinear model gives a testable tradeoff: root placement recovers from a 15° stationary release that the side layout does not, while allowing greater first overshoot in the matched release tests.

Measuring torque against angle and recording real release motion would test whether those predictions hold for the prototype; this rigid-layer study does not establish individual-piece stability.