• Prize list A: Prizes offered for
solving teaching challenges (volumes I to V)

• Prize list B: Prizes offered for special
achievements (volumes I to V)

• Prize list C: Prizes offered around the strand
model (volume VI)

• Prize list D: Prizes offered for graphics and
animations

•
Prize winners for previous challenges

The text contains over 1700 challenges in physics and maths. Some solutions are not included. For challenge solutions of particular didactic or research value I offer special prizes. The first good solution I receive will be added to the free physics textbook, together with the sender's name. Partial prizes may also be awarded. To apply for a prize, mail a solution to christoph@motionmountain.net. Prize list B is for special achievements. Several challenges from prize lists C and F could lead to publications or form chapters in a PhD thesis.

### Prize list F: Favourite strand challenges and prizes offered for their
solution:

Challenge F 3: Determine, using the tangle model, the mass ratio between the neutrinos and the electron. Prize: 500 euro – and publish the solution!

Challenge F 4: Determine, using the tangle model, a better approximation for the fine structure constant than 1/126. Prize: 500 euro – and publish the solution!

Challenge F 5: Determine how the probability of belt-trick-like rotation for a polymer-tethered ball depends on the radius of the ball and the number of polymer chains. First, look at this video. Then, imagine that the belts are exchanged by two parallel cylindrical polymer chains, and that there are not only 4, but 2n of them; also imagine that the central buckle is a ball - either spherical or ellipsoidal. In the strand model, the probability for the belt trick is related to the absolute mass of the elementary particles. So solving this polymer problem in detail will also advance particle physics. The solver will receive a special prize plus an amount of at least 500 euro – and the solution should be published!

Challenge F 6: Falsify any of the predictions due to strands given here, with experiments or other arguments. The prize will be substantial. Please write to set terms together.

Challenge F 7: Prove, clarify or disprove this conjecture: no visualization of SU(2) or SU(3) without tethers is possible in 3 dimensions. Prize: 50 euro.

Challenge F 8: Prove, clarify or disprove this conjecture: no non-equivalent visualization of SU(3) to the one with three strands is possible in 3 dimensions. (A triangular belt buckle with three attached belts attached with hinges does not count as "inequivalent".) Prize: 50 euro.

Challenge F 9: Prove, clarify or disprove this conjecture: no visualization of SU(n) with strands for n > 3 is possible in 3 dimensions. (This would have profound consequences for physics, including the lack of GUTs.) Prize: 50 euro.

Challenge F 10: Prove, clarify or disprove this conjecture: no visualization of SU(n) with n > 3 is possible in 3 dimensions, of any kind. Prize: 50 euro.

Challenge F 11: Prove, clarify or disprove this conjecture: the rational tangle classification is mathematically complete and leaves no room for additional fermions or bosons. (This clarifies the options for dark matter.) Prize: 150 euro.

Challenge F 12: Prove, clarify or disprove this conjecture: the rational tangle classification is mathematically complete and leaves no room for additional defects in space that are neither fermions nor bosons. (This clarifies the options for dark energy.) Prize: 50 euro.

### PRIZE LIST A: Prizes offered for solving teaching challenges (volumes I
to V)

Challenge A 2: The 'tangles inside a sphere' problem. This problem combines topology, statistics and geometry. Estimate the number of topologically different tangles that can fit into a sphere of given volume, with the assumption that every strand, though flexible, has constant diameter. A glass sphere of radius R contains n strands of diameter d (d<R), all starting and ending on the surface (at 2n given and fixed points distributed over the surface). How many topologically different tangles can be formed, under the condition that the diameter d has the largest possible value for a given n? [Note: in mathematics, one distinguishes trivial, composed, rational, locally knotted and prime tangles. The problem asks for the number of possible tangles that are composed or prime. Locally knotted tangles may be left out of the counting; rational tangles do not count as topologically different in this problem.] Prize: 1000 euro – and publish the solution! One attempt so far.

Challenge A 3: Classical Lagrangian for waves. Use the relation for the errors in angular frequency and time for wave packets, dw dt > 1/2, to show that the classical action for a wave is bounded below. Find the precise bound by assuming that the initial and final points for which the action is determined must themselves obey the wave packet relation. Prize: 100 euro – and publish the solution!

Challenge A 4: Rotation in special relativity. Make a movie of a sphere/football with relativistic speed and relativistic rotation speed. Show the strange effects that appear. Prize: 100 euro – and publish the solution! Half an attempt so far.

Challenge A 5: The simplest unsolved knot problem. Imagine an ideally wobbly rope, that is, a rope that has the same radius everywhere, but whose curvature can be changed as one prefers. Tie a trefoil knot into the rope. By how much do the ends of the rope get nearer? In 2011, there are only numerical estimates for the answer: about 10.1 diameters. There is no formula giving the number 10.1 yet - can you find one? Alternatively, solve the following problem: what is the rope length of a closed trefoil knot? Also in this case, only numerical values are known - about 16.33 rope diameters - but no exact formula. Prize: 500 euro from me, plus eternal fame :-) – thus publish the solution! Two attempts so far.

Challenge A 6: Extending the belt trick to spin 3/2. The Dirac belt trick simulates the behaviour of a spin 1/2 particle. What is the construction for a composed spin 3/2 particle? For an elementary spin 3/2 particle? Prize: 50 euro. One attempt so far.

Challenge A 7: Taking a combined photograph of a rainbow, similar to the one by Stefan Zeiger, but including a third segment with the ultraviolet picture. Prize: 50 euro.

Challenge A 8: Explain the problems in performing a Bohm-type experiment with two nuclei that are first near each other and then separated. Prize: 50 euro.

Challenge A 9: Convince Adobe to eliminate an ugly and old bug: in Adobe Reader, horizontal line thicknesses are shown on screen irregularly in wrong, usually exaggerated thicknesses. That makes tables look horrible. The bug has been noted and communicated to Adobe by many since version 1 of the program; it is still present in version XI. This one will receive a special reward, because even several Adobe engineers gave up on the topic in the past. The reward is 50 euro plus a paper copy of the 6 volumes signed by the author, with a special thank you letter in the name of all graphics artists worldwide. Three attempts so far.

Challenge A 10: Supply valuable details on the mathematics of tree growth: the laws about their proportions, the height of their trunks, as derived from the principle of minimum effort, etc. Prize: 50 euro. Two attempts so far.

Challenge A 12: Provide figures of the most important topological defects: twirls, hedgehogs, etc. Prize: 10 euro.

Challenge A 15: Supply beautiful and clear photos for wave interference, polarization, diffraction, refraction, wave damping, and wave dispersion. Prize: 20 euro each.

Challenge A 16: Write an article about the site and book in a newspaper or journal. Prize: a paper edition.

Challenge A 17: Find a research
literature reference for the difficulties encountered when trying to
observe *classical* tachyons, i.e. the difficulties encountered when
a *classical* tachyon (assuming it exists) flies across your field
of view. (There is a better reference than Recami, somewhere, I know it
from hearsay. There is also a website somewhere, I saw it and lost it in
2014.) Prize: 50 euro. One attempt so far.

Challenge A 18: Take and send in pictures or films about striking examples of motion in biology or in any other field - with permission of use. Prize: 50 euro.

Challenge A 19: Propose good solutions to challenges marked as "ny". Prize: 10 euro each.

Challenge A 20: Propose counter-examples to maximum force, maximum power, or maximum mass flow. Prize: 200 euro for valid ones, 10 euro for false, but good ones.

### PRIZE LIST B: Prizes offered for special achievements (volumes I to V)

Challenge B 1: Propose substantial improvements to the text. I will reward your input in a way that is comparable to the other rewards listed here.

Challenge B 3: Translate untranslated parts of the text into Spanish, Italian, German, French or any other language. Click here for details on how to do it and then let me know. I will find a way to reward your efforts appropriately.

### PRIZE LIST C: Prizes offered for challenges about the strand
model (volume VI)

Challenge C 2: Determine the probability of twists (Reidemeister Type I moves) for a chiral tangle made of two or three strands. Show that the square of the probability is 1/137.036. Prize: 1000 euro – and publish the solution!

Challenge C 3: Determine, from the strand model, how the mass of the W and Z bosons run with energy. Prize: 500 euro – and publish the solution!

Challenge C 6: Deduce, using the
strand model, an *analytical* approximation for particle mass ratios or
particle masses. The approximation could be based on ropelength or on the W/Z
mass ratio calculation - but does not have to be. Prize: 1000 euro –
and publish the solution!

Challenge C 7: Find, using the
strand model, an *analytical* approximation for probabilities of core
shape changes. Prize: 1000 euro – and publish the
solution!

Challenge C 9: Describe, for the strand model, the relation between strand distance and the modulus of the wave function. Prize: 50 euro.

Challenge C 10: Deduce in a simple way, for the strand model or for textbook quantum mechanics, Schwinger's quantum action principle from the quantum of action. Prize: 100 euro.

Challenge C 11: Find, for the strand model, a simple visualization of the weak mixing angle, also called the 'Weinberg angle'. Prize: 100 euro.

Challenge C 13: Describe, for the strand model, the precise phase choices for a crossing that lead to SU(3) invariance. Prize: 200 euro.

Challenge C 14: Visualize, for the strand model, the relation between SU(3) in the harmonic oscillator and SU(3) due to the third Reidemeister move. Prize: 200 euro.

Challenge C 15: Check and estimate the six basic lepton tangles and their ropelength. Prize: a surprise.

Challenge C 17: Confirm or invalidate the new W and Z tangles defined with the help of strands. Prize: a surprise.

### PRIZE LIST D: Prizes offered for graphics and animations

Challenge D 1: Produce animations about the three Reidemeister moves. Prize: 50 euro.

Challenge D 2: Produce animations about the fluctuating tangles of quarks and leptons. Prize: 100 euro.

Challenge D 3: Produce an animation about the fundamental principle. Prize: 50 euro.

Challenge D 4: Produce animations about the gauge interactions, to be discussed together. Prize: to be decided.

### PRIZE WINNERS for previous challenges

Solved challenge A 1: The parking problem. Find the minimum number of times one has to drive backwards and forwards to leave from a parking space, when the available space and the geometry of the car are given. Look up the details by searching for 'car parking' in the book index. After several attempts by others, the prize has been awarded to Daniel Hawkins.

Solved challenge A 11: Provide a research literature reference for the diffusive speed of light in the solar interior. Prize awarded to Zach Espiritu.

Solved challenge A 13: Send in a quality CAD drawing of a Foucault gyroscope. Prize also awarded to Zach Espiritu.

Solved challenge A 14: Take and send in a beautiful and clear photograph of how a candle flame reacts to a rubbed comb or, better, of how it splits in a high electric field. Prize awarded to Shubham Das and Rakesh Kumar.

Solved challenge A 20: Produce an animation that is unique throughout the world, one that shows clearly how two belts behave as fermions under exchange, complementing the now standard visualisation that a belt behaves like a fermion under rotations. Together with the usual belt trick, the animation would thus visualize the spin-statistics theorem for spin 1/2. The ideas and the original physical behaviour to be shown are straightforward, but the graphic work will take a bit of patience. The resulting animation would extend the applet shown at http://www.gregegan.net/APPLETS/21/21.html to a situation with two belt buckles and four belts. If you are interested, get in touch with me. One option could be to start with the software at www.cs.indiana.edu/~hansona/quatvis/Belt-Trick/index.html and then expanding it. The resulting animation will be unique world-wide, and I would build it into the text. The prize goes to Antonio Martos for his wonderful animation.

*

Solved challenge C 4: Extrapolate the measured up and down quark masses to Planck energy, using the standard model. Solved by Shun Zhou.

Solved challenge C 8: Formulate, for the strand model, a simple relation between strand diameter and the Planck time. Solved in the new edition of volume VI.

Solved challenge C 12: Find, for the strand model, a compelling deduction of the gluon tangles. Solved in the new edition of volume VI.

Solved challenge C 16: Calculate or estimate the ropelength of the (corrected) Higgs tangle (volume VI). Solved by Eric Rawdon and Rob Sharein.

*

Solved challenge F 1: Discover the correct tangles, in volume VI, for the leptons. Solved in 2018 by CS.

Solved challenge F 2: Draw a 3d picture in eps format of space-time curvature and allow me to use it in the text. Solved in 2018 by Farooq Ahmad Bhat.