This is a solution to the Guzzle©
30 October 2024
1. Convention:
N - North passing through the North Pole (Arctic)
S - South passing through the South Pole (Antarctic)
E - East, W - West, F - Front, B - Back
N S E W F and B are axes fixed in space, with axis F pointing towards you when you are holding the globe with the axis N up. Please note that there are eight octants namely: N-F-W, N-F-E, N-W-B, N-E-B, S-F-W, S-F-E, S-W-B, and S-E-B, which are fixed in space.
Clockwise rotation and counterclockwise rotation: If you position a normal right handed screw on any axis, with the screw pointed toward the center of the earth, clockwise rotation is the direction which will drive the screw into the earth. Clockwise rotation is considered positive and one quarter (90 degrees) clockwise rotation about each of the six axes is indicated by: N S E W F B respectively. (2S) indicates two quarter rotations about S. One quarter counterclockwise rotation is indicated by: N’ S’ E’ W’ F’ B’ respectively.
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2. Preliminaries.
The Guzzle in the form of a ball has eight spherical parts, rotatable about the center of the ball (also translatable due to this) about the six axes N S E W F and B. When the spherical parts are properly positioned and oriented the Guzzle represents a world Globe. In general the varied positions and orientations of the spherical parts will not represent the world Globe familiar to us. To bring back all the spherical parts and their orientations to their respective positions to represent the world Globe is a challenge and is the object of the puzzle.
Before attempting to solve the puzzle, it is advantageous to familiarize ourselves with each of the spherical parts. The proper positions and orientations of the spherical parts when the puzzle is solved are as follows: the equator divides the globe into the northern hemisphere and the southern hemisphere. It is also easy to see that the equator divides the two continents namely Africa and South America into southern and northern parts. The northern part of Africa is adjoining Europe, and the northern part of South America adjoins North America.
The configuration to represent the world Globe is as follows. In the Northern Hemisphere, the four spherical parts (traversing in the east direction) consist of a) Africa-Europe [AE], b) Asia-India-Japan [AIJ], c) Alaska-Hawaii [AH] together with North Pacific Ocean and some western part of mainland North America, d) North America-Greenland [NAG] together with the northern part of South America above the Equator. In the Southern Hemisphere, the four spherical parts consist of e) Southern-Africa-Madagascar [SAM] together with a quarter of Antarctica located below AE, f) Australia-New Guinea [ANG] together with another quarter of Antarctica located below AIJ, g) New Zealand_South Pacific Ocean [NZS] together with another quarter of Antarctica located below AH, h) Southern Part of South America [SA] together with another quarter of Antarctica located below NAG.
3. Solution for the Northern Hemisphere.
Hold the globe so that the part Africa-Europe (AE) faces you and to the top and left of center. The part of the arctic should be at the top towards the north (N). This position fixes AE with respect to the N-F-W octant. The part containing Asia-India-Japan (AIJ) needs to be on the top and right. If this is not the current position find AIJ and rotate so that it goes to the octant defined by W-S-B, without changing the position of AE, after which follow the rotations indicted below:
- Perform the rotations: F-(2S)-F’
- Now see that this brings the part AIJ next to the part AE as required. If AIJ is also in the proper orientation then we have AIJ positioned and oriented correctly and the arctic part of AIJ is next to the arctic part of the Africa-Europe. If AIJ is NOT properly oriented, repeat the following sequence of rotations until AIJ is correctly oriented:
E’-(2S)-F-(2S)-F’.
We now have the two parts AE and AIJ of the northern hemisphere in the proper positions and orientations.
- Next we need to bring the part containing Alaska-Hawaii [AI] next to the part AIJ. To do this, rotate the whole globe so that AIJ is to the left and the arctic part is on top near N. This fixes the part AIJ with respect to N-F-W octant. Find the Alaska-Hawaii part and move it to the W_S_B octant making sure that the parts positioned so far (AE and AIJ) have not changed their positions.. In this new position repeat the steps a) and b) above and confirm that the Alaska-Hawaii part moves to the correct position and orientation. Similarly, bring the North America-Greenland part next to the Alaska-Hawaii part.
Confirm that all the above steps position and orient all the parts of the Northern Hemisphere.
4. Solution for the Southern Hemisphere.
The solution for the Southern Hemisphere has to be obtained without disturbing the positions and orientations of the Northern hemisphere parts obtained in section 3 above. Hence, the solution for the Southern Hemisphere is more involved and several cases need to be considered.
Turn the globe upside down so that the arctic is now near the S axis. Observe the locations of the four Antarctic parts of the Southern Hemisphere parts looking down from the axis N. The various possible cases are described below.
Case a): Two of the Antarctic parts are next to each other near the N axis, and
when held such that they are on the right of the center, the other two Antarctic parts are also next to each other but are near the equator on the left side near axis W as shown in the figure below:
For this case, perform the following sequence of rotations: F-E-N-E’-N’-F’
This will result in all the four Antarctic parts on top of the globe near axis N as shown in the figure below:
Case a) 1.
In the figure above we find that the two parts below W E need to be relocated. Perform the following sequence of rotations:
E-N-E’-N’-E’-F-(2E)-N’-E’-N’-E-N-E’-F’
to solve the puzzle.
Case b): Two of the Antarctic parts are next to each other near the N axis as shown in the figure below. When held such that they are on the right of the center, the other two Antarctic parts are near the equator on the left side but on opposite ends one near axis F and the other near axis B as shown in the figure below.
In this case perform the following sequence of rotations: E-N-E’-N’-E’-F-E-F’
This sequence of rotations brings all the four Antarctic parts near axis N. Also two of the diagonally opposite parts will be properly located and oriented, and the other two are diagonally juxtaposed with respect to their locations.
In this case the following sequence of rotations need to be performed to solve the puzzle: F-E-N’-E’-N’-E-N-E’-F’-E-N-E’-N’-E’-F-E-F’
The above sequence, which is quite involved (contains a total of seventeen rotations),
Case c): Only one of the Antarctic octant is on the top right position when looking down the globe from N as shown below:
In this case the Antarctic part on the top left is near the equator and near axis W, the Antarctic part on the bottom right is also near the equator and near axis E, and the Antarctic Part below and to the left is also near the equator and axis F. In this case perform the following sequence of rotations :
E-(2N)-E’-N’-E-N’-E’
The above sequence positions the southern hemisphere parts such that at least two parts are positioned and oriented correctly and the other two parts are adjacently juxtaposed as shown in the figure below:
To bring these two adjacently juxtaposed parts to the proper positions we need to execute the following sequence of rotations to solve the puzzle::
E-N-E’-N’-E’-F-(2E)-N’-E’-N’-E-N-E’-F’
Case d): None of the Antarctic parts are on top. The Antarctic parts on the left are together near the equator and W whereas the Antarctic parts on the right are away from each other near the equator as seen in the figure below.
In this case perform the following sequence of rotations: F-(E-N-E’-N’)2–F’
After the above sequence we will see that all the Antarctic parts are on the top near axis N as required. The above set of rotations will bring two adjacent partsin the proper order and the other two adjacent parts are juxtaposed to each other as in the figure below:
In this case perform the following sequence of rotations to solve the puzzle:
E-N-E’-N’-E’-F-(2E)-N’-E’-N’-E-N--E’-F’
Case e): Only one of the Antarctic parts is on the top near the north (N) as in the figure below looking down from the axis N. This is on the part on the left and below. The other three Antarctic parts are along the equator as seen in the figure below.
For this case perform the following sequence of rotations::
E-N-E’-N-E-(2N)-E’
Case f): Two of the Antarctic parts are diagonally opposite each other and these two parts are properly positioned and oriented as shown in the figure below:
We can place all the parts in the proper positions by executing the following sequence of rotations: N-E-N’-W-N-E’-N’-W’
Repeat the above rotation sequence until all the parts are in the correct positions. When all the octants are in the correct positions we see that all of them are not oriented properly and two of the Antarctic parts are on the right adjacent to each other and the other two Antarctic parts are near the equator and on opposite sides as shown in the figure below.
In this case perform the following sequence of rotations: E-N-E’-N’-E’-F-E-F’
The above sequence of rotations leads to the following figure wherein two of the Antarctic parts are near N and are diagonally placed and the other two Antarctic parts are near the equator, one near F and another near W, as shown in the figure below:
which gives the globe orientation for the following sequence of rotations:
F’-E-N-E’-N’-E’-F-E
An additional rotation on the equator may be needed to solve the puzzle. The above sequence puts all the parts in the correct positions and orientations thus solving the puzzle.
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