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I was studying up on some pretty archaic methods for testing intelligence and I came across one called The Wheel of Judas. It is said that if you were able to fully solve the wheel that you were a danger to the kingdom and were sentenced to a lifetime of servitude outside of the capital (pretty brutal punishment for being smart). It was pretty confusing at first, but over time I managed to solve it; I am wondering if the community can do the same?


The Wheel

The wheel can seem a bit overwhelming at first; I heavily recommend you take your time when studying it.

The Wheel of Judas

The way it works is quite complicated so I will break it down.


Rotation

The wheel will rotate a total of $4$ times from it's starting position above. Each rotation is $90^\circ$ to the right at a time; however, you may think that everything will rotate but the inventor was quite a genius! The wheel is broken down into five categories, which function in different ways:

  • Outer-Numbers: These rotate in order (see below).
  • Inner-Numbers: Simply swap with each rotation.
  • Vowels and Word Plates: These pieces of the wheel are stationary and do not move.

  • Mathematical Containers: These rotate through the order of operations (see below).


The Outer and Inner Numbers

As stated above, this portion of the wheel is pretty simple. With each rotation, the outer-numbers rotate $90^\circ$ to the right; the inner numbers simply swap back and forth. For example:

$\require{AMScd}$ \begin{CD} 1 @>>> 3\\ @AAA 5 \leftrightarrow 6 @VVV\\ 4 @<<< 2 \end{CD}


The Mathematical Containers

This is where it gets tricky, especially once the rules are laid out. The mathematical containers house the consonants of the alphabet and rotate through the order of operations (PEMDAS) with each rotation of the wheel. For example:

$\require{AMScd}$ \begin{CD} * @>>> /\\ @AAA @VVV\\ + @<<< - \end{CD}


The Vowels and Word Plates

These sections are stationary and do not move with rotation. The cool thing about the vowel chamber is the rules that apply to it:

  • The letter $a$ must always be used.
  • One additional vowel may be used (only one per word).
    • The additional vowel can come before or after the $a$.

The word plates are where your chosen words go. They are located under each number. Once a word is placed within a plate, it cannot be moved to a different location. This means that a word placed at number $1$ at startup, will be at number $4$ on the first rotation, $2$ on the second, and so on. To clarify:

  • You can choose a number for all six words; however, once all six words have been found and the outer and inner numbers begin to rotate, the words remain at those chosen locations on the wheel.

The Objective

Find $6$ words that meet the following criteria:

  • Each word uses the letter $A$.
  • Each word must be at least three letters.
  • Each word has at least one consonant.
    • Each consonant remains on it's side of the vowel when created.
    • Each consonant follows the order of operations in it's placement.
  • Each word connects to it's currently designated partner via the following:
    • The first letter of $a$ is equal to the last letter of $b$.
    • It's placement on the board is linked to another:
    • For example:
    • $1=2, 2=6, 3=4, 4=3, 5=1, 6=5$

    • You may choose the pairings, but no number can link to another number that has already been linked. Another possible linking is (though this will make it too easy):
    • $1=2, 2=3, 3=4, 4=5, 5=6, 6=1$

    • $+50$ reputation bonus to someone who solves with a unique linking of numbers, that isn't $1,2,3,4,5,6$ or $6,5,4,3,2,1$.

A Correct Word Pair Example

Note: The words given in this example will not count in a correct answer, no matter their placement.

Assuming the wheel is in its current state, a correct word pairing that meets the basic criteria would be chant and guac. The reason behind this is:

  • In CHANT:

    • The vowel $A$ is used.
    • The consonant $C$ (a multiplication) comes before $H$ (an addition) and both come before the vowel.
    • The consonant $N$ (a division) comes before $T$ (a subtraction) and both come after the vowel.
  • In GUAC:

    • The wheel has been rotated.
    • The vowel $A$ is used.
    • Only one extra vowel ($U$) was added.
    • The consonant $G$ (an addition) comes before the vowel.
    • The consonant $C$ (a multiplication) comes after the vowel and is the first letter in CHANT.

Question

Since there are many possibly combinations due to there being thousands of words available to choose from, the first technically correct answer will win. If the answer happens to be the answer I also arrived at, then I will award a $+50$ reputation bounty when I am able.

What are six words that meet all given criteria for all four rotations?


Clarification Section

If you have anything you need clarification on, please let me know so that I can clear it up prior to your attempts at an answer. I tried to be as thorough as possible, but this can seem confusing and quite a daunting puzzle in my opinion.

  • The mathematical containers are rotated during the word search process; once per word with the first rotation occurring after the discovery of word one.
  • The mathematical containers were drawn in left to right, top to bottom order on purpose. The rotation of the containers is described correctly above. Don’t think of it as MDAS then SMDA, it rotates the available operations in the way depicted. Maybe someone better at phrasing can assist with the representation here?
  • Each word is assigned to a plate below a number, the numbers are paired together; for example $1$ and $2$, $2$ and $6$, The full pairing example is supplied above.
  • In the example given chant is at position $1$ and guac at position $2$; this means that position $6$ must end with $G$ somehow.
  • When building your word, order of operations is considered on each side of the vowel separately. Read the example above carefully.

Hints

This is a rather difficult puzzle, and I had a really difficult time creating it. Here's an idea that may help you:

A simple approach (but difficult to accomplish) is to find words that jump back and forth.

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  • $\begingroup$ What do you mean by "It's current partner is determined by it's number in which...". What were the numbers for guac and chant in your example. Also, does order of operations matter when separated by a vowel (I would think chant would be wrong since a division comes after an addition otherwise) $\endgroup$ – JGibbers Oct 10 '18 at 21:32
  • $\begingroup$ Since this is an ancient test, should we use an ancient language? :) By the way, there are closer to 500,000 English words in the current Websters Third New International Unabridged dictionary. Did you mean to imply that non-English words are acceptable? $\endgroup$ – SteveV Oct 10 '18 at 21:46
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    $\begingroup$ Okay, so with the mathematical sections rotating, you end up with I think: the two on the left must be used in top-down fashion (if used), and before the vowel, then the right two again in top-down only after the vowel. Ex: in CHANT, MA had to happen top down, thus MA and not AM, and could not mix with DS. In GUAC, post rotation, your options become: AS (in that order only, on the left) and MD (again, only that order, on the right). Is this correct? $\endgroup$ – Dorrulf Oct 10 '18 at 23:24
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    $\begingroup$ Hmm, I'll try one more time for now. Ex: Find CHANT, then rotate the consonants only (still 90 deg), find GUAC, rot cons, continue until 6 words are found. Now, rotate the outer wheel (the numbers) 4 times, and I assume test that the 6 words still follow all the rules each time (including rotating the consonants in the previous fashion). If the 6 words worked each time, you have it @PerpetualJ? $\endgroup$ – Dorrulf Oct 10 '18 at 23:59
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    $\begingroup$ @gabbo1092 You get to choose where each word goes before rotation of the outer and inner numbers; once that begins, the words have to remain in place. $\endgroup$ – PerpetualJ Oct 11 '18 at 14:23
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Here is a simple solution for the pairing $1=2$, $2=1$, $3=4$, $4=3$, $5=6$, $6=5$:

With that pairing if you have three sets of words that start and end with the same letters it will work after the rotations because the rotation change what two words are compared.

First word I chose before the word plates rotate is bait, placed in spot 1 (before numbers rotate), after first word rotation tab is made and placed in spot two, next rotation make tac in spot 3, then coat in spot 4, then dear in spot 5, then rad in spot 6.

Bit of a cop out with this pairing. Still looking for a more complex pairing solution.

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