Changes in c185851: updating pctX page with table of contents and a proper image

				haas/spring2026/common/projects/images/pctX_borrows.jpg
			
          Binary files /dev/null and b/haas/spring2026/common/projects/images/pctX_borrows.jpg differ

				haas/spring2026/common/projects/pctX.md
			
          @@ -1,4 +1,49 @@

          -# PROJECT: PRACTICING CRITICAL THINKING (pctX)

          +# PROJECT: PRACTICING CRITICAL THINKING (PCTX)

          +

          +# TABLE OF CONTENTS

          +

          +  * [OBJECTIVE](#objective)

          +  * [BACKGROUND](#background)

          +    * [INVESTIGATION/LOGIC METHODS](#investigation/logic-methods)

          +    * [MATH PREPARATION](#math-preparation)

          +    * [LONG DIVISION](#long-division)

          +  * [LETTER DIVISION: AN EXAMPLE](#letter-division-an-example)

          +    * [FINDING K (AND J AND L AND O AS WELL)](#finding-k)

          +    * [FINDING OUR ZERO VALUE](#finding-our-zero-value)

          +    * [FINDING V AND G](#finding-v-and-g)

          +    * [FINDING M AND DISCOVERING P](#finding-m-and-discovering-p)

          +  * [GETTING STARTED](#getting-started)

          +  * [PROCESS](#process)

          +  * [A NOTE ON NUMBER BASES](#a-note-on-number-bases)

          +  * [YOUR SUBMISSION](#your-submission)

          +    * [SUBMISSION FOR REGULAR LETTER DIVISION](#submission-for-regular-letter-division)

          +    * [SUBMISSION FOR SOLVE4 LETTER DIVISION](#submission-for-solve4-letter-division)

          +  * [PUZZLE KEY](#puzzle-key)

          +  * [PUZZLE SOLUTION](#puzzle-solution)

          +  * [VERIFICATION](#verification)

          +    * [REGULAR PUZZLE](#regular-puzzle)

          +    * [EXAMPLE FOR REGULAR PUZZLE](#example-for-regular-puzzle)

          +  * [SOLVE4 PUZZLE](#solve4-puzzle)

          +  * [WALKTHROUGH VIDEOS](#walkthrough-videos)

          +  * [STRATEGIES](#strategies)

          +    * [LEFT EDGE](#left-edge)

          +  * [DETERMINE BORROWS AND TAKES](#determine-borrows-and-takes)

          +  * [TOP IS KNOWN GREATER THAN](#top-is-known-greater-than)

          +  * [TOP IS KNOWN LESS THAN](#top-is-known-less-than)

          +  * [RIGHT EDGE](#right-edge)

          +  * [LOOK FOR ZERO AND GREATEST SYMBOL CANDIDATES](#look-for-zero-and-greatest-symbol-candidates)

          +  * [PROCESS OF ELIMINATION](#process-of-elimination)

          +  * [DOUBLING](#doubling)

          +  * [NEXT-TO HINTS](#next-to-hints)

          +  * [MORE NEXT-TO HINTS](#more-next-to-hints)

          +  * [SUBTRACT BY GREATEST DIGIT, GET INCREMENT](#subtract-by-greatest-digit,-get-increment)

          +  * [SUBTRACT BY KNOWN OFFSET FROM GREATEST DIGIT, GET OFFSET INCREMENT](#subtract-by-known-offset-from-greatest-digit,-get-offset-increment)

          +  * [DIVISOR/MULTIPLICATION RELATIONS](#divisor/multiplication-relations)

          +  * [INVERTED SUBTRACTION PAIRS](#inverted-subtraction-pairs)

          +  * [INVERTED SUBTRACTION RELATIONAL PAIRS](#inverted-subtraction-relational-pairs)

          +  * [SINGLE-LETTER SUBTRAHEND DETERMINATION](#single-letter-subtrahend-determination)

          +  * [CHECKING YOUR RESULTS](#checking-your-results)

          +  * [SUBMISSION](#submission)

           ## OBJECTIVE

          @@ -155,7 +200,7 @@ basic long division process before  attempting a letter division problem.

           So,  be  sure  to  try  your  hand at  a  few  practice  problems  before

           proceeding.

          -## LETTER DIVISION: an example

          +## LETTER DIVISION AN EXAMPLE

           Following  will be  a sample  letter division  problem, and  a documented

           solution of  it, much as you  will be doing  for this project (and  to be

          @@ -337,7 +382,9 @@ We  cannot instantly  proceed to  the next  subtraction in  as obvious  a

           progression,  as  we'll need  more  information  on the  various  letters

           involved.

          -### Finding K (and J and L and O as well)

          +### FINDING K

          +

          +... and J and L and O as well.

           However, looking at  the puzzle, I'm interested in seeing  if we can find

           any obvious  examples of 0.  You know, letter  minus same letter  sort of

          @@ -489,7 +536,9 @@ right is borrowing from it (because R  < B), we have something like this:

           Can't really do much  more with it at this point, but  it is important to

           know to help us identify the borrows needing to happen.

          -### Finding our zero value (R and B)

          +### FINDING OUR ZERO VALUE

          +

          +(R and B)

           Why don't we  go ahead and find  0? If you look in  the subtraction above

           the bottom one, we have another  "letter minus same letter" scenario, and

          @@ -549,7 +598,7 @@ Relational chains can look as follows now:

           Basically just down to V, G, P, and M.

          -### Finding V and G

          +### FINDING V AND G

           And  I think  we have  the means  to find  V: notice  the second  to last

           subtraction, the  "LKBKV". You know  where we get that  from? Multiplying

          @@ -600,10 +649,10 @@ V = {       2                      }

           Relational chains can look as follows now:

          -  * R < B < V < G << M << O < L < J < K

          -  * R < B < V < G << P << O < L < J < K

          +  * `R < B < V < G << M << O < L < J < K`

          +  * `R < B < V < G << P << O < L < J < K`

          -### Finding M and discovering P

          +### FINDING M AND DISCOVERING P

           And then  there were 2. We  really just need to  find M, or P,  and we're

           done. And since there are no 'P'  values in the puzzle, we need to target

          @@ -659,13 +708,13 @@ V = {       2                      }

           Relational chains can look as follows now:

          -  * R < B < V < G < P < M < O < L < J < K

          +  * `R < B < V < G < P < M < O < L < J < K`

           I wasn't able to show  it as well in text on the wiki,  but I also made a

           point to  mark up each subtraction  to show whether a  borrow occurred or

           not:

          -{{ :undefined:borrows.jpg?400 |}}

          +![marked borrows and takes](images/pctX_borrows.jpg)

           To  be  sure,  there are  likely  MANY,  MANY  ways  to arrive  at  these

           conclusions.  What is  important  is being  observant, performing  little

          @@ -767,7 +816,7 @@ may have to do (certainly on lower difficulty levels of puzzle).

           ## YOUR SUBMISSION

          -### SUBMISSION FOR STANDARD-STYLE LETTER DIVISION

          +### SUBMISSION FOR REGULAR LETTER DIVISION

           If your puzzle  was provided with a quotient and  remainder (and contains

           no question marks in the puzzle proper), you have a regular puzzle.

          @@ -781,7 +830,7 @@ The files you will want to submit include:

           Your solution MUST be of a form so that, if given to another person, they

           can follow your steps and have an understanding of the decisions made.

          -### SUBMISSION FOR SOLVE4-STYLE LETTER DIVISION

          +### SUBMISSION FOR SOLVE4 LETTER DIVISION

           The point behind a "solve4" puzzle is to also determine the `QUOTIENT` and `REMAINDER`, in addition to the key.

          @@ -1234,7 +1283,7 @@ Also:

           In either case of  T being odd or even, we can  eliminate half the values

           (if T is even, it cannot be any odd values, not in an even base).

          -## NEXT-TO HINTS

          +## NEXT TO HINTS

           Sometimes you may be treated to left-most clues like this:

          @@ -1250,7 +1299,7 @@ Notice how J-F equals nothing? That tells us the following things:

             * F is exactly one value less than J (written: F < J)

             * K is LESS THAN G and T (K has to borrow to make J-F=0 versus the 1 it would otherwise be).

          -## MORE NEXT-TO HINTS

          +## MORE NEXT TO HINTS

           What really pays off is when we have a scenario like this:

          @@ -1542,7 +1591,7 @@ So, rolling out with known borrow-takes:

               C     X

           ```

          -## SINGLE-LETTER SUBTRAHEND DETERMINATION

          +## SINGLE LETTER SUBTRAHEND DETERMINATION

           NOTE: Mostly useful for the  "solve4" category of letter division puzzles

           providing a conducive scenario to utilize this strategy.

...	...	@@ -1,4 +1,49 @@
1		-# PROJECT: PRACTICING CRITICAL THINKING (pctX)
	1	+# PROJECT: PRACTICING CRITICAL THINKING (PCTX)
	2	+
	3	+# TABLE OF CONTENTS
	4	+
	5	+ * [OBJECTIVE](#objective)
	6	+ * [BACKGROUND](#background)
	7	+ * [INVESTIGATION/LOGIC METHODS](#investigation/logic-methods)
	8	+ * [MATH PREPARATION](#math-preparation)
	9	+ * [LONG DIVISION](#long-division)
	10	+ * [LETTER DIVISION: AN EXAMPLE](#letter-division-an-example)
	11	+ * [FINDING K (AND J AND L AND O AS WELL)](#finding-k)
	12	+ * [FINDING OUR ZERO VALUE](#finding-our-zero-value)
	13	+ * [FINDING V AND G](#finding-v-and-g)
	14	+ * [FINDING M AND DISCOVERING P](#finding-m-and-discovering-p)
	15	+ * [GETTING STARTED](#getting-started)
	16	+ * [PROCESS](#process)
	17	+ * [A NOTE ON NUMBER BASES](#a-note-on-number-bases)
	18	+ * [YOUR SUBMISSION](#your-submission)
	19	+ * [SUBMISSION FOR REGULAR LETTER DIVISION](#submission-for-regular-letter-division)
	20	+ * [SUBMISSION FOR SOLVE4 LETTER DIVISION](#submission-for-solve4-letter-division)
	21	+ * [PUZZLE KEY](#puzzle-key)
	22	+ * [PUZZLE SOLUTION](#puzzle-solution)
	23	+ * [VERIFICATION](#verification)
	24	+ * [REGULAR PUZZLE](#regular-puzzle)
	25	+ * [EXAMPLE FOR REGULAR PUZZLE](#example-for-regular-puzzle)
	26	+ * [SOLVE4 PUZZLE](#solve4-puzzle)
	27	+ * [WALKTHROUGH VIDEOS](#walkthrough-videos)
	28	+ * [STRATEGIES](#strategies)
	29	+ * [LEFT EDGE](#left-edge)
	30	+ * [DETERMINE BORROWS AND TAKES](#determine-borrows-and-takes)
	31	+ * [TOP IS KNOWN GREATER THAN](#top-is-known-greater-than)
	32	+ * [TOP IS KNOWN LESS THAN](#top-is-known-less-than)
	33	+ * [RIGHT EDGE](#right-edge)
	34	+ * [LOOK FOR ZERO AND GREATEST SYMBOL CANDIDATES](#look-for-zero-and-greatest-symbol-candidates)
	35	+ * [PROCESS OF ELIMINATION](#process-of-elimination)
	36	+ * [DOUBLING](#doubling)
	37	+ * [NEXT-TO HINTS](#next-to-hints)
	38	+ * [MORE NEXT-TO HINTS](#more-next-to-hints)
	39	+ * [SUBTRACT BY GREATEST DIGIT, GET INCREMENT](#subtract-by-greatest-digit,-get-increment)
	40	+ * [SUBTRACT BY KNOWN OFFSET FROM GREATEST DIGIT, GET OFFSET INCREMENT](#subtract-by-known-offset-from-greatest-digit,-get-offset-increment)
	41	+ * [DIVISOR/MULTIPLICATION RELATIONS](#divisor/multiplication-relations)
	42	+ * [INVERTED SUBTRACTION PAIRS](#inverted-subtraction-pairs)
	43	+ * [INVERTED SUBTRACTION RELATIONAL PAIRS](#inverted-subtraction-relational-pairs)
	44	+ * [SINGLE-LETTER SUBTRAHEND DETERMINATION](#single-letter-subtrahend-determination)
	45	+ * [CHECKING YOUR RESULTS](#checking-your-results)
	46	+ * [SUBMISSION](#submission)
2	47
3	48	## OBJECTIVE
4	49
...	...	@@ -155,7 +200,7 @@ basic long division process before attempting a letter division problem.
155	200	So, be sure to try your hand at a few practice problems before
156	201	proceeding.
157	202
158		-## LETTER DIVISION: an example
	203	+## LETTER DIVISION AN EXAMPLE
159	204
160	205	Following will be a sample letter division problem, and a documented
161	206	solution of it, much as you will be doing for this project (and to be
...	...	@@ -337,7 +382,9 @@ We cannot instantly proceed to the next subtraction in as obvious a
337	382	progression, as we'll need more information on the various letters
338	383	involved.
339	384
340		-### Finding K (and J and L and O as well)
	385	+### FINDING K
	386	+
	387	+... and J and L and O as well.
341	388
342	389	However, looking at the puzzle, I'm interested in seeing if we can find
343	390	any obvious examples of 0. You know, letter minus same letter sort of
...	...	@@ -489,7 +536,9 @@ right is borrowing from it (because R < B), we have something like this:
489	536	Can't really do much more with it at this point, but it is important to
490	537	know to help us identify the borrows needing to happen.
491	538
492		-### Finding our zero value (R and B)
	539	+### FINDING OUR ZERO VALUE
	540	+
	541	+(R and B)
493	542
494	543	Why don't we go ahead and find 0? If you look in the subtraction above
495	544	the bottom one, we have another "letter minus same letter" scenario, and
...	...	@@ -549,7 +598,7 @@ Relational chains can look as follows now:
549	598
550	599	Basically just down to V, G, P, and M.
551	600
552		-### Finding V and G
	601	+### FINDING V AND G
553	602
554	603	And I think we have the means to find V: notice the second to last
555	604	subtraction, the "LKBKV". You know where we get that from? Multiplying
...	...	@@ -600,10 +649,10 @@ V = { 2 }
600	649
601	650	Relational chains can look as follows now:
602	651
603		- * R < B < V < G << M << O < L < J < K
604		- * R < B < V < G << P << O < L < J < K
	652	+ * `R < B < V < G << M << O < L < J < K`
	653	+ * `R < B < V < G << P << O < L < J < K`
605	654
606		-### Finding M and discovering P
	655	+### FINDING M AND DISCOVERING P
607	656
608	657	And then there were 2. We really just need to find M, or P, and we're
609	658	done. And since there are no 'P' values in the puzzle, we need to target
...	...	@@ -659,13 +708,13 @@ V = { 2 }
659	708
660	709	Relational chains can look as follows now:
661	710
662		- * R < B < V < G < P < M < O < L < J < K
	711	+ * `R < B < V < G < P < M < O < L < J < K`
663	712
664	713	I wasn't able to show it as well in text on the wiki, but I also made a
665	714	point to mark up each subtraction to show whether a borrow occurred or
666	715	not:
667	716
668		-{{ :undefined:borrows.jpg?400 \|}}
	717	+![marked borrows and takes](images/pctX_borrows.jpg)
669	718
670	719	To be sure, there are likely MANY, MANY ways to arrive at these
671	720	conclusions. What is important is being observant, performing little
...	...	@@ -767,7 +816,7 @@ may have to do (certainly on lower difficulty levels of puzzle).
767	816
768	817	## YOUR SUBMISSION
769	818
770		-### SUBMISSION FOR STANDARD-STYLE LETTER DIVISION
	819	+### SUBMISSION FOR REGULAR LETTER DIVISION
771	820
772	821	If your puzzle was provided with a quotient and remainder (and contains
773	822	no question marks in the puzzle proper), you have a regular puzzle.
...	...	@@ -781,7 +830,7 @@ The files you will want to submit include:
781	830	Your solution MUST be of a form so that, if given to another person, they
782	831	can follow your steps and have an understanding of the decisions made.
783	832
784		-### SUBMISSION FOR SOLVE4-STYLE LETTER DIVISION
	833	+### SUBMISSION FOR SOLVE4 LETTER DIVISION
785	834
786	835	The point behind a "solve4" puzzle is to also determine the `QUOTIENT` and `REMAINDER`, in addition to the key.
787	836
...	...	@@ -1234,7 +1283,7 @@ Also:
1234	1283	In either case of T being odd or even, we can eliminate half the values
1235	1284	(if T is even, it cannot be any odd values, not in an even base).
1236	1285
1237		-## NEXT-TO HINTS
	1286	+## NEXT TO HINTS
1238	1287
1239	1288	Sometimes you may be treated to left-most clues like this:
1240	1289
...	...	@@ -1250,7 +1299,7 @@ Notice how J-F equals nothing? That tells us the following things:
1250	1299	* F is exactly one value less than J (written: F < J)
1251	1300	* K is LESS THAN G and T (K has to borrow to make J-F=0 versus the 1 it would otherwise be).
1252	1301
1253		-## MORE NEXT-TO HINTS
	1302	+## MORE NEXT TO HINTS
1254	1303
1255	1304	What really pays off is when we have a scenario like this:
1256	1305
...	...	@@ -1542,7 +1591,7 @@ So, rolling out with known borrow-takes:
1542	1591	C X
1543	1592	```
1544	1593
1545		-## SINGLE-LETTER SUBTRAHEND DETERMINATION
	1594	+## SINGLE LETTER SUBTRAHEND DETERMINATION
1546	1595
1547	1596	NOTE: Mostly useful for the "solve4" category of letter division puzzles
1548	1597	providing a conducive scenario to utilize this strategy.