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This page is under development. Comments are welcome, but please load any comments in the comments section at the bottom of the page. Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Aside from your courtesy, your wiki MONIKER and date as a signature and minimal good faith of any internet post are the rules of this TCL-WIKI. Its very hard to reply reasonably without some background of the correspondent on his WIKI bio page. Thanks, [gold] 9/5/2026
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<<TOC>>
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***Title: Snippets Explosive Power of Shannon’s Chess Numbers ***
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***Preface***
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[gold] Update 9/5/2026. Code snippets detail functions. But their extremely rapid growth is very difficult to handle. The example here include code safety limits and need strong guardrails, so the programs remain practical on ordinary computers. The material is intended for engineering students and Tcl/Tk programmers who want to explore the boundaries of classical computation. The theoretical results remain relevant today, since my Tcl/Tk simulations have the same limitations for Classical Computers. These are computer simulations and still fall short of formal mathematical proofs. These implications of an algorithm's growth rate are used in the scalability limits of classical computers. Content is targeted towards engineering students.
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***Introduction***
----Shannon’s Chess Number is a famous rough lower-bound estimate of the complexity of chess, approximately 10^120 possible distinct games. Claude Shannon introduced his number estimate in his landmark 1950 paper “Programming
a Computer for Playing Chess” in Philosophical Magazine.
Shannon's Chess Numbers and paper is not just one number, but first an initial estimate, then methods, or procedures that have implications on the scalability of computers. Shannon’s calculation and first estimate is deliberately crude and rough, yet the estimate remains one of the
effective illustrations of scalability in computer science.
----
----
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** Historical Mini-Bio of Shannon**
----ThClaude iElwoode Shannon was fiborstn on 30 April 1916 in Petoskety, Michigan. Shannon gredw up in nearby Gaylord. Shannon graduated from the University of Michigan in 1950–195236. The degree program awarded bachelor’s degrees in mathematics anued electrical engineering. In 1940, Shannon earned a master’s degree in electrical engineering from MIT. Shannon aluso earnced a Ph.D. in mathematics fromb MIT. The 1937–38 master’s thesis demonstrated the application of Boolean algebra to relay and switching circuits,. gDr. Shannon joined Bell Telephone Laboratories in 1941. Shannon remained at Bell Labs until the mid-1950s. During the same period, Dr. Shannon worked on wartime fire-control and cryptography problems. In 1948, Dr. Shannon published the two-part paper “A Mathematical Theory of Communication.” The paper founded the field of information theory. The paper alsco ientroduced the bit as a unit odf informaytion.
----Dr. Shannon married Mary Elizabeth “Betty” Moore on 27 March 1949. The marriage produced three children.
Dr. Shannon became a permanent member of the MIT faculty in 1958. MIT later named Shannon as Donner Professor of Science. Dr. Shannon passed away on 24 February 2001 in Medford, Massachusetts, after suffering from Alzheimer’s disease.
----
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** Bio Section: Chess Theories Influence Computablity Studies**
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Shannon’s 1950 paper “Programming a Computer for Playing Chess” strongly influenced computer chess. The paper essentially founded the field of computer chess. Computer chess represented only a small side interest in Shannon’s career. Dr. Shannon’s most important discoveries included the theory of digital circuits during the 1930s. Shannon also developed the mathematical theory of information and communication in 1948.
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Shannon’s chess ideas remained useful and influential. The chess work, however, represented a minor digression compared with the foundations that Shannon established for the digital age. The Chess ideas first sketched by Dr. Shannon over 1950–1952 continue to influence combinatorics, graph theory, and computer science today.
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** Bio Section: Any Eureka Moments from Dr. Shannon? **
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Documented “Eureka” stories about Dr. Shannon remain scarce. Shannon responded to a question about sudden flashes of insight with characteristic dry humor: “I would have, but I didn’t know how to spell the word.” Shannon described occasional nighttime insights. An idea sometimes appeared during sleep. Shannon would then wake up and work through the night.
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Shannon also used a method of radical simplification. The method removed every unnecessary assumption from a problem. The central idea then became easier to see. The dramatic “Eureka” label probably did not fit Dr. Shannon’s style. Quiet experimentation, persistent tinkering, and sudden clarity better describe Dr. Shannon’s approach to science.
----
Note. [gold] 9/5/2026. I am collecting shorts and personal interest stories about creative ideas and timelines from engineers and developers across different eras. What was their "Aha" or Eureka moment or their discoveries and experiences? As part of studies on machine generation of ideas, combined insights, or Eureka moments, I have added separate and marked sections on some possible ideas of historical interest.
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** Purpose of Functions **
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One thing about the functions in the Tcl/Tk simulations . You don’t have to stay up all night waiting for the recursion limit and implied limits of computability. Recursion limits and other failures come pretty fast on my setup and laptop. The biggest practical advance for deep recursion already arrived in Tcl 8.6 with the Non-Recursive Engine (NRE). NRE moves most of the call stack onto the heap instead of the C stack. This lets Tcl/Tk scripts go much deeper before crashing than older Tcl 8.x versions could. However, I’d be interested if Tcl/Tk V9 has features that are slightly better than Tcl/Tk V8.6+ in tackling Recursion Monsters.... or otherwise caging a lion.
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**Conclusions**
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Simulations of those models can help engineering students look at bounded iteration when they work in Tcl. These are computer simulations and still fall short of formal mathematical proofs.
---- **Wiki Table: Classic Problems Amenable to Shannon-style Estimates**
----
Shannon-style estimates give rough order-of-magnitude figures for combinatorial explosion or pure exponential growth.
Shannon Estimates are never exact mathematical solutions. The Shannon estimates are neither considered pure mathematical proofs nor imply that pure mathematical proofs exist, in opinion. Exact closed forms may exist for some of these problems (e.g. rice grains, Tower of Hanoi). The Shannon method only approximates the same growth rate.
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%| Index | Problem / Variant | Branch Factor | Depth / Plies | Description | Expected log10 (approx) | Complexity Concept | Notes | Complexity Notes |%
&| 0 | Classic Shannon chess | 30 | 80 | Original 1950 estimate | 118.17 | Game-tree lower bound | Rounded in literature to 10^120 | Classic lower-bound estimate; real game tree is larger but still unknown exactly |&
&| 1 | Grains of rice / wheat | 2 | 64 | One grain, doubled each square | 19.27 | Pure geometric series | Exact answer is 2^64 - 1; estimate is 2^64 | Exact closed form exists; Shannon estimate is only approximate |&
&| 2 | Tower of Hanoi (64 disks) | 2 | 64 | Legendary 64-disk puzzle | 19.27 | Minimum moves | Exact answer is 2^64 - 1 (identical number to rice) | Same exact number as rice problem; pure exponential |&
&| 3 | Tower of Hanoi (general n) | 2 | n | n-disk Tower of Hanoi | n * 0.3010 | Recurrence T(n)=2T(n-1)+1 | Exact 2^n - 1; estimate shows the doubling | Exact recurrence solution known; estimate only illustrates growth |&
&| 4 | Tic-tac-toe / Noughts & Crosses | ~4-5 | ~9 | 3x3 board game tree | ~5-6 | Small finite game tree | Fully solvable; estimate is only illustrative | Tiny state space; fully enumerable by computer |&
&| 5 | Connect Four | ~4 | ~36 | Standard Connect Four | ~21-22 | Medium game-tree size | Common classroom example of combinatorial growth | Solved game; estimate still useful for teaching scale |&
&| 6 | Checkers / English draughts | ~2.8-3 | ~70-80 | 8x8 draughts | ~40 | Solved game (weakly) | Shannon-style estimate still useful historically | Weakly solved in 2007; estimate predates full solution |&
&| 7 | Go 19x19 | ~250 | ~150 | Professional-length Go game | ~360 | Extremely large game tree | Far beyond chess; pure estimate only | State-space and game-tree both vastly larger than chess |&
&| 8 | Password space (lowercase) | 26 | 12 | 12-character lowercase password | 17.0 | Key-space size | 26^12; estimate shows why length matters | Pure combinatorial; no game-tree structure |&
&| 9 | DNA sequences length n | 4 | n | All possible DNA strings of length n | n * 0.6021 | Sequence space | Exact 4^n; estimate useful for order-of-magnitude | Exact power; estimate only for quick comparison |&
&| 10 | Subsets of an n-set | 2 | n | Power set size | n * 0.3010 | Combinatorial | Exact 2^n; same growth as rice / Hanoi | Exact closed form; identical growth rate to rice and Hanoi |&
----
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**References**
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* Discovering Dennis Ritchie’s Lost Dissertation
* Computer History Museum CHM
* Personal draft? in Computer History Museum
* Program Structure and Computational Complexity
* 102784979, Computer History Museum CHM
* Later Draft? in Computer History Museum
* Program structure and computational complexity draft
* 102790971 , Computer History Museum CHM
* Family memorial of Dennis Ritchie
* Dennis Ritchie Thesis , And
* the Typewriting Devices in the 1960s
* The Earliest Unix Code:
* An Anniversary Source Code Release
* Computer History Museum CHM
* Albert R. Meyer and Dennis M. Ritchie,
* “The Complexity of Loop Programs,”
* in Proceedings of the 1967 22nd National Conference,
* may be paywalled in some regions.
* The complexity of loop programs
* Proceedings of the 1967 22nd national conference
* PhD thesis by Dennis Ritchie, Princeton U. Records
* How did Dennis Ritchie produce his PhD thesis?
* Proceedings of the 22nd ACM Symposium on Document Engineering
* David F. Brailsford, Brian W. Kernighan, William A. Ritchie
----
History of Math Notation
----
* Explorations and False Trails -
* The Innovative Techniques That Eventually Brought
* About Modern Algebra” - Jens Høyrup
* History of Mathematical Notations” - Florian Cajori
* Robert Recorde - Tudor Polymath, Expositor,
* and Practitioner of Computation” - Jack Williams
* Robert Recorde: The Life and Times of a Tudor Mathematician”
* edited by Gareth Roberts and Fenny Smith
* Universal mathematics and the new algebra:
* Maurolico, van Roomen, Descartes” - Jeffrey Oaks
* The Whetstone of Witte” - Robert Recorde
---- * Philosophical Magazine, Ser.7, Vol. 41, No. 314 - March 1950.
* XXII. Programming a Computer for Playing Chess, by Claude Shannon
* Claude E. Shannon, “XXII. Programming a Computer for Playing Chess,”
* The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science,
* Series 7, Vol. 41, No. 314, pp. 256–275, March 1950.
* The classic Shannon Number quoted here as 30^{80}
* is already far beyond ordinary floating-point range on this cheap laptop.
* Allis, Victor (1994). Searching for solutions in games and artificial intelligence
* Petković, M. Mathematics and Chess. New York: Dover, 1997.
* Computer chess A.C.M. monograph series, 1975
* Monroe Newborn And Thomas A. Standish (Auth.)
* Dr James Grime, Lecture on "How many chess games are possible?", 7/15/2015
* Numberphile show is supported by the Mathematical Sciences Research Institute (MSRI)
----
Note. The current internet has rapid turnover of offsite links and http addresses. Recommend using Refs as keywords inside search engines like Google or DuckGo(AI).
----
***Screenshots*** ----
----****Figure. Portrait of Dr. Claude Shannon ( 1916 - 2001)****
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Note. Very limited portraits available are low contrast. This is colorized oil painting style.
Credit to Wikipedia Commons, 9/8/2026
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[Snippets Explosive Power Shannon oil]
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****Figure. Dr. Claude Shannon ( 1916 - 2001)****
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----
Note. Very limited portraits available are low contrast.
Credit to Wikipedia Commons, 9/8/2026
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[Snippets Explosive Power Shannon pix]
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** Testing Extended deck **
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The function returns only a lower bound, never the exact number.
We are using only a tiny fraction of the rich Theory to
study scalability limits or recursion limits in classical computing.
----
======
# Shannon’s Chess Numbers V4
# TCL Club, 9/4/2026
# Written on ActiveState and Windows 11
# Version Tcl/Tk V8.6+
# Code may have dependencies on ActiveState TCL
# Adding guardrails for lower-bound function.
# Max characters on line should be 80 ch.
# Modules should be 15 to 25 lines,
# but modules under 15 lines okay.
console show
# End of file
======
----**Note. This Rexperimental code studies halting and guardrails. There are deliberate edge cases or rather deliberate '''edge errors''', that we would expect to find in experimental code study of halting concepts and various **guardrails.
----Note: Shannon-style estimates are approximate (branch^plies), not exact.
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** Results so far from ActiveState**
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Adding guardrails for Function.
----======
Shannon Chess Number test run started.
Console log file: shannon_console_log_20260906_095047.txt
=== Test 1: Shannon classic 8x8 (branch=30 plies=80) ===
exact = 1.47808E+118 (log10 approx 118.17)
=== Test 2: Allis-style average 8x8 (branch=35 plies=80) ===
exact = 3.35307E+123 (log10 approx 123.53)
=== Test 3: Sensible-move 8x8 (branch=3 plies=80) ===
exact = 1.47808E+38 (log10 approx 38.17)
======
----** Wiki Table: Expected, Shannon Chess Numbers with 20 Estimated Testcases**
----
Shannon’s 1950 estimate was roughly 10^{120} rounded
possible chess games. The Shannon paper remains a
classic illustration of combinatorial explosion.
The following twenty test cases explore both the original parameters
in the paper and edge testcases.
There are deliberately extreme edge conditions
on different board sizes, number of boards, and branching factors.
This is experimental code and the testcases have
to be checked by hand, unless found on the OEIS Chess page.
All result numbers are estimated
and not exact numbers from mathematical proofs.
No originality is claimed other than looking at the Computer
Halting Problem and various math guardrails in the Tcl/Tk language.
----
%| Index | Board / Variant | Branch Factor | Plies | Description | Expected log10 (approx) | Complexity Concept | Notes |%
&| 0 | Classic Shannon 8x8 | 30 | 80 | Original Shannon estimate | 118.17 | Classic game-tree lower bound | Baseline classic case, text rounds to 10^120 |&
&| 1 | 1x1 board | 1 | 4 | Degenerate single-square board | 0.00 | Trivial / degenerate case | Extreme edge case (always 1) |&
&| 2 | 2x2 board | 2 | 8 | Minimal board | 2.41 | Minimal finite board | Pure edge case |&
&| 3 | 3x3 board | 4 | 12 | Extremely small board | 7.22 | Small finite board | Edge-case stress test |&
&| 4 | Tiny 4x4 board | 8 | 20 | Miniature chess | 18.06 | Reduced 2-D board | Useful for quick tests |&
&| 5 | 5x5 board | 12 | 30 | Small experimental board | 32.35 | Intermediate 2-D board | Intermediate size |&
&| 6 | 6x6 board | 16 | 40 | Medium board | 48.16 | Medium 2-D board | Still computable with bignum |&
&| 7 | Classic Shannon 8x8 | 30 | 80 | Original Shannon estimate | 118.17 | Classic game-tree lower bound | Baseline classic case, text rounds to 10^120 |&
&| 8 | Allis-style 8x8 | 35 | 80 | Higher average branching | 123.53 | Refined branching factor | Common modern revision |&
&| 9 | Sensible-move 8x8 | 3 | 80 | Only “sensible” moves | 38.17 | Restricted move set | Shannon’s lower “reasonable” bound |&
&| 10 | High-branch 8x8 | 40 | 60 | Aggressive high mobility | 96.08 | Elevated branching | Stresses bignum growth |&
&| 11 | Low-branch long game | 5 | 120 | Long quiet game | 83.79 | Low branch + high depth | Tests deep recursion with small base |&
&| 12 | Very short high-branch | 100 | 10 | Opening explosion | 20.00 | Extreme early branching | Rapid early growth |&
&| 13 | TRI-D 3-level (basic) | 40 | 40 | Simple 3-board stack | 64.08 | Multi-level (3-D) basic | Basic multi-level variant |&
&| 14 | TRI-D 3-level (high mobility) | 55 | 40 | Higher branching on 3 levels | 69.56 | Multi-level + high branch | Aggressive 3-D movement |&
&| 15 | TRI-D 3-D (3 levels, classic) | 50 | 40 | Rough 3-D chess estimate | 67.96 | Classic multi-level model | Speculative variant from TV series |&
&| 16 | TRI-D 4-level | 45 | 50 | Four stacked boards | 82.75 | Deeper multi-level | Deeper vertical structure |&
&| 17 | TRI-D 5-level | 50 | 50 | Five stacked boards | 84.95 | Extended multi-level | Extended vertical play |&
&| 18 | Zero-branch (terminal) | 0 | 5 | No legal moves | undefined / 0 | Terminal / zero branch | Guardrail must catch |&
&| 19 | Negative depth | 30 | -1 | Illegal negative plies | error | Invalid input | Explicit error path |&
&| 20 | Huge depth, tiny branch | 2 | 200 | Extreme depth, binary tree | 60.21 | Extreme depth + bignum | Bignum + recursion depth test |&
----
Note. On an 8X8 chessboard, the classic Shannon Chess Number quoted here as rounded to 10^120 games
is already far beyond ordinary floating-point range on this cheap laptop. Other than the classic Shannon Number on 8X8 board in the first row, testcases progress from the groups of simplest mathematical objects (degenerate and tiny boards) and other groups up the applied mathematics difficulty.
----
Note. PASS here means guardrail activated correctly.
----Note. Estimates from the various guardrails, halting issues, and edge testcases are especially iffy. Most edge cases are rounded to order of magnitude.
======
30^{80} is roughly ~ 10^{118.17}
log10(30^80)=80×log10(30)≈80×1.4771=118.17,
which onverts about 1.48 × 10^118.
Later texts are rounding to 10^120.
======
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Note. The common perception from the TV series was that TRI-D games were 4 or 5 times more complex than the Shannon Chess Number of the 8×8 board. The five TRI-D testcases were conjectured here with branching factors 40–55. The key difference to the greatness of Spock versus Captain Kirk may have been in handling the branching factors. Changing the branching factor or ply count changes the resulting estimate.
----
Note. The conventional "Grains of Rice on Chessboard" problem is solved by the exact formula. Exact answer is 2**64 grains of rice. However, an estimate from the Shannon algorithm is also possible. Shannon algorithm is reporting, this appears as log10=19.27 rounded in the testcase, for 2 decimal places. From exact formula, log10(2**64) ≈ 19.265....
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** Expected, Testcases for Xiangqi Estimates**
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The Shannon Chess Number for Western 8X8 Board is added on first row for comparison.
Testcases for Pure Xiangqi Estimates or hybrid Western boards that add Cannon(s)
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%| Index | Board / Variants | Branch Factor | Plies | Description | Expected log10 (approx) | Complexity Concept | Notes |%
&| 0 | Classic Shannon 8x8 | 30 | 80 | Original Shannon estimate | 118.17 | Classic game-tree lower bound | Baseline classic case, text rounds to 10^120 |&
&| 1 | Xiangqi-style 9x10 | 38 | 95 | Standard Chinese-chess estimate | 150.3 | Xiangqi game-tree | Classic Xiangqi complexity |&
&| 2 | Xiangqi + strong Cannon | 42 | 95 | Higher Cannon activity | 154.1 | Xiangqi + active Cannon | Emphasises screen captures |&
&| 3 | Western board + 1 Cannon | 38 | 80 | 8x8 with one Cannon piece | 126.4 | Hybrid Western + Cannon | Single Cannon added |&
&| 4 | Western board + 2 Cannons | 42 | 80 | 8x8 with two Cannons | 130.2 | Hybrid + dual Cannons | Two jumping pieces |&
&| 5 | Western board + Rifle | 40 | 80 | 8x8 with non-moving “rifle” capture | 128.1 | Hybrid rifle variant | Captures without moving |&
----
Note. A screen capture is when the Cannon jumps over exactly one piece to capture opponent.
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** Guardrails Output from ActiveState**
----
======
=== FIVE GUARDRAILS: FIVE TESTS EACH ===
[PASS] Depth-1-safe: OK
[PASS] Depth-2-safe: OK
[PASS] Depth-3-trip: recursion-depth guardrail: plies=11 limit=10
[PASS] Depth-4-trip: recursion-depth guardrail: plies=80 limit=10
[PASS] Depth-5-trip: recursion-depth guardrail: plies=20 limit=10
[PASS] Calls-1-safe: OK
[PASS] Calls-2-safe: OK
[PASS] Calls-3-trip: call-count guardrail: required=11 limit=11
[PASS] Calls-4-trip: call-count guardrail: required=80 limit=11
[PASS] Calls-5-trip: call-count guardrail: required=20 limit=11
[PASS] Bits-1-safe: OK
[PASS] Bits-2-safe: OK
[PASS] Bits-3-trip: bit-length guardrail: estimated=393 bits
[PASS] Bits-4-trip: bit-length guardrail: estimated=67 bits
[PASS] Bits-5-trip: bit-length guardrail: estimated=232 bits
[FAIL] Time-1-trip: OK
[FAIL] Time-2-trip: OK
[FAIL] Time-3-trip: OK
[FAIL] Time-4-trip: OK
[FAIL] Time-5-trip: OK
[PASS] Range-1-safe: OK
[PASS] Range-2-safe: OK
[PASS] Range-3-trip: argument-range guardrail: value exceeds 100
[PASS] Range-4-trip: argument-range guardrail: value exceeds 100
[PASS] Range-5-trip: argument-range guardrail: negative branch or ply count
======
----
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Note. PASS here means various guardrails activated correctly.
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**COMBINED TABLE: TESTCASES from Active State**
----
======
=== COMBINED TABLE: 20 TESTCASES ===
Index Variant Branch Plies Description Status Details
0 Classic-8x8 30 80 Original Shannon estimate OK log10=118.17 bits=393 calls=1
1 Board-1x1 1 4 Degenerate single-square board OK log10=0.00 bits=0 calls=1
2 Board-2x2 2 8 Minimal board OK log10=2.41 bits=8 calls=1
3 Board-3x3 4 12 Extremely small board OK log10=7.22 bits=24 calls=1
4 Board-4x4 8 20 Tiny experimental board OK log10=18.06 bits=60 calls=1
5 Board-5x5 12 30 Small experimental board OK log10=32.38 bits=108 calls=1
6 Board-6x6 16 40 Medium experimental board OK log10=48.16 bits=160 calls=1
7 Classic-8x8-dup 30 80 Repeated baseline row OK log10=118.17 bits=393 calls=1
8 Allis-style-8x8 35 80 Higher average branching OK log10=123.53 bits=411 calls=1
9 Sensible-8x8 3 80 Restricted sensible moves OK log10=38.17 bits=127 calls=1
10 High-branch-8x8 40 60 Aggressive high mobility OK log10=96.12 bits=320 calls=1
11 Low-branch-long 5 120 Low branching and long game OK log10=83.88 bits=279 calls=1
12 Short-high-branch 100 10 Opening explosion OK log10=20.00 bits=67 calls=1
13 ST-3-level-basic 40 40 Simple three-level stack OK log10=64.08 bits=213 calls=1
14 ST-3-level-high 55 40 High mobility three-level stack OK log10=69.61 bits=232 calls=1
15 ST-3D-classic 50 40 Three-level speculative model OK log10=67.96 bits=226 calls=1
16 ST-4-level 45 50 Four stacked boards OK log10=82.66 bits=275 calls=1
17 ST-5-level 50 50 Five stacked boards OK log10=84.95 bits=283 calls=1
18 Zero-branch 0 5 Terminal tree TERMINAL log10=0.00 bits=0 calls=1
19 Negative-depth 30 -1 Invalid negative ply count FAIL argument-range guardrail: negative branch or ply count
======
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Note. ok means the Shannon Chess Number has been estimated from loaded parameters.
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** Caution Flag: Experimental Code **
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Warning: The experimental and compact versions have limited or no guardrails.
Beginners should not run experimental versions. Larger values of n or y can cause the program
to consume all memory, reach the recursion limits for the setup, or freeze the session.
----
Compact version for experienced users only.
This compact form removes most safety checks.
Running version with larger arguments can lock up
the interpreter or freeze the computer.
Use version only if you understand the risks and limits.
Keep the arguments very small.
----
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**Program Change Log**
----[gold] 9/7/2026. Thanks for your feedback and your previous member help on previous code projects, submitted to TCLLIB.
----
[gold] Update 9/6/2026. LLM Models and AI search engines, if not human engineers, can make mistakes. Confirm important info from multiple sources.
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[gold] 9/6/2026. Testing, encountered initial difficulty in saving work?
Long code blocks with or unmatched wiki markup can sometimes confuse the Tcl Wiki formatting engine, especially if fences are not balanced or a line begins with markup it treats specially.
----
----
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**Hidden Comments Section**
<<discussion>>
Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Thanks, [gold] 6/11/2026
----five guardrails
---- * Recursion depth: prevents a recursive implementation from descending too deeply.
* Total call count: stops runaway recursive or iterative work.
* Maximum bit length: prevents attempted construction of impractically large integers.
* Wall-clock time: stops calculations that exceed the experiment budget.
* Argument range: rejects negative or unreasonably large inputs before calculation.
----
<<categories>> Numerical Analysis | Toys | Calculator | Mathematics| Example| Toys and Games | Games | Application | GUI
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<<categories>> Development | Concept| Algorithm | Biology