Chess Mania

Numerical Chess: Revealing the Numerical Structure of the Game
NUMERICAL CHESS – CURRENT STATE 2026
Episode 26 – Structural Analysis [10] – The Hidden Weight of Structural Connections
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Numerical Chess — Credo
Chess positions are numerical structures.
Pieces create connections that generate measurable influence.
The game unfolds through the transformation of these numbers.
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In the previous episode we learned how to count structural connections:
• IC – Invasive Connections (red)
• AC – Advancing Connections (green)
• BC – Base Connections (blue)
However, simple counting does not yet reveal their true structural force.
Not every connection has the same impact on the position.
A connection that invades the opponent’s territory is clearly more powerful than one that merely supports from the base.
Therefore Numerical Chess introduces two new instruments:
1. IC equivalents (ICeq) — measuring structural pressure
2. TP9 — measuring local tactical tension
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1. IC Equivalents (ICeq)
To measure structural pressure more accurately, all connections are converted into a single common unit:
IC equivalents (ICeq).
The weighting formula is:
ICeq = IC + 0.5(AC−IC) + 0.25(BC−AC)
This reflects the structural hierarchy of connections.
Connection Structural role Weight
IC invasive pressure 1.00
AC advancing support 0.50
BC base stability 0.25
Thus:
• IC contribute fully
• AC contribute half
• BC contribute a quarter
ICeq therefore measures the true structural pressure of a zone.
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2. Example: Structural Evaluation of a Position
White
Zone 1
IC = 4
AC = 7
BC = 4
ICeq = 4.75
Zone 2
IC = 7
AC = 5
BC = 2
ICeq = 6.00
Total White structural pressure:
10.75 ICeq
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Black
Zone 1
IC = 4
AC = 6
BC = 4
ICeq = 4.50
Zone 2
IC = 6
AC = 4
BC = 2
ICeq = 5.00
Total Black structural pressure:
9.50 ICeq
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Structural Interpretation
White has a clear but moderate structural advantage.
10.75 − 9.5 = +1.25
Within Numerical Chess this means:
White's network of connections exerts slightly greater structural pressure across the board.
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3. Tactical Pressure: TP9
While ICeq measures global structural force, tactical conflicts normally arise in smaller areas of the board.
For this reason Numerical Chess introduces TP9, which measures tactical pressure in the 9-zones.
The formula is simple:
TP9 = IC + 0.5 ⋅ AC / 9
Only IC and AC are considered here, because BC connections are too remote to generate immediate tactical pressure.
The result is a tactical pressure value.
Typical TP9 values look like:
TP9 Meaning
0.20 – 0.40 low tactical tension
0.40 – 0.60 moderate tension
0.60 – 0.70 strong tension
> 0.70 highly critical zone

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4. TP9 in the Current Position
White
TP9 ≈ 0.61
Black
TP9 ≈ 0.67
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Tactical Interpretation
Both sides have significant tactical pressure in the local zones.
However:
Black’s value is slightly higher, approaching the critical threshold of 0.70.
This indicates that tactical friction in the local area currently favors Black.
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5. Structural vs Tactical Layers
The position therefore shows an interesting dual structure.
Layer Advantage
Structural layer (ICeq) White
Tactical layer (TP9) Black
White controls the overall structural network, while Black generates slightly stronger local tactical pressure.
Such configurations often lead to dynamically balanced middlegames, where:
• structure tries to stabilize the position,
• tactics attempt to disturb the balance.
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6. The New Insight
Episode 26 introduces an important principle of Numerical Chess:
Connections must not only be counted, they must be weighted.
By converting IC, AC and BC into IC equivalents, structural pressure becomes measurable.
And by measuring TP9, the system also reveals local tactical tension.
Together these two values allow us to observe something remarkable:
A chess position behaves like a field of interacting numerical forces, where structural stability and tactical pressure constantly influence each other.
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Numerical Chess
Exploring the hidden numerical structure of chess.

NUMERICAL CHESS – CURRENT STATE 2026

Author’s Note Before Episode 26

From Counting Connections to Measuring Their Strength

In the previous episodes of Numerical Chess, we learned how to count structural connections:
• IC – Invasive Connections
• AC – Advancing Connections
• BC – Base Connections
This already allows us to describe the numerical structure of a position surprisingly well.
As Episode 25 demonstrated, even without seeing the board it is possible to infer:
• structural balance
• strategic tendencies
• initiative vs. long-term stability
• the approximate phase of the game
However, simple counting has a limitation.
Not all connections have the same practical power.
An Invasive Connection (IC) that penetrates the opponent’s territory is usually more influential than an Advancing Connection (AC), and an AC is typically more active than a Base Connection (BC) that mainly stabilizes the structure from behind.
In other words:
Connections differ not only in type, but also in strength.
This leads to the next step in the development of Numerical Chess.
Instead of only counting connections, we begin to measure their effective influence.
Episode 26 therefore introduces two important concepts:
IC equivalents (ICeq)
A weighted measure that converts IC, AC and BC into a single structural pressure value.
TP9 (Tactical Pressure in the 9-zones)
A compact indicator that measures local tactical tension.
Together these two instruments allow us to distinguish between:
• global structural pressure
• local tactical pressure
From this point onward, Numerical Chess moves from describing structure to measuring structural force.
And this reveals something remarkable:
A chess position is not just a static arrangement of pieces.
It is a dynamic network of numerical influences, where structural depth and tactical pressure constantly interact.
Episode 26 begins this next step.

Numerical Chess: Revealing the Numerical Structure of the Game
NUMERICAL CHESS – CURRENT STATE 2026
Episode 25 – Structural Analysis [9] – What Can We Know Without Seeing the Position?
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Numerical Chess — Credo
Chess positions are numerical structures.
Pieces create connections that generate measurable influence.
The game unfolds through the transformation of these numbers.
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Take a look at the attached simplified NC algorithm table.
This table contains only data about the number and zonal distribution of
• IC (Invasive Connections)
• AC (Advancing Connections)
• BC (Base Connections)
extracted from a particular game position.
Now imagine the following challenge:
You are asked to describe this position, evaluate it, and suggest what both sides should do — without seeing the board.
Is that even possible?
First of all, it must be said:
There are certainly millions of positions that could display similar data regarding the number and zonal distribution of IC, AC, and BC. Visually, many of these positions would look very different from one another.
However, it is still possible to describe the numerical structure and to provide a structural evaluation of the entire group of positions.
Everything beyond that becomes partly speculative — for example, which pieces are actually on the board.
Since the author of these lines had to see the position in order to enter the data into the table, his faithful AI collaborator AI-Sha was asked about the matter.
At first, she was shown only the table — not the position itself.
Most notably, her structural analysis — the most important part of the evaluation — proved to be astonishingly accurate and, among the millions of possible positions with similar connection data, almost perfectly applicable to the actual position.
Here are some excerpts from her observations:
“This is a very interesting exercise, because the table actually contains far more positional information than one might expect without seeing the board. Structural connections alone already reveal a surprising amount about the phase, balance of forces, and strategic tendencies of the position.”
Game Phase
“Late middlegame or endgame.”
In fact, it was an endgame.
But what kind of endgame?
AI-Sha proposed several possibilities, one of which turned out to be remarkably precise:
“Rook + 3–4 pawns each.”
Indeed, in the actual position White has a rook and four pawns, while Black has a rook and three pawns.
No other pieces are present except the kings — whose approximate placement AI-Sha could also infer from the connection structure.
General Evaluation of the Position Group
AI-Sha’s answer:
“Approximately balanced, with a slight advantage for White.”
This was fundamentally correct. In the actual position, White has only one concrete winning motif; otherwise the game should end in a draw.
Naturally, AI-Sha also pointed out structural asymmetries in AC and BC, which generally give White better long-term prospects. This is because AC and BC can, in principle, be transformed into IC, and White has significantly better chances in this respect.
White needs this advantage, of course, because his current IC values are still well below the tactically critical thresholds that AI-Sha knows and has partly calculated herself.
She wrote much more, and from a purely structural perspective, almost everything was correct.
Afterwards, the actual position was finally shown to her, and she was asked whether anything about it surprised her.
She replied:
“Yes — to some extent it is a surprise, but interestingly most structural conclusions remain remarkably consistent with the real position.”
And further:
“The real surprise is not the board. The real surprise is how much was already correct without seeing it.
From the connection numbers alone we correctly inferred:
✔ structural imbalance
✔ initiative vs. structural depth
✔ moderate pressure level
✔ absence of decisive tactical overload
✔ stable but asymmetric structure
That is remarkably strong information for a purely numerical reading.”
She added:
“This position is actually an excellent teaching example. It shows that structural connections alone already encode the strategic DNA of a position.”
And also:
“Even without the board, we could already see:
• who has the initiative
• who has the deeper structure
• which side benefits from time
• which side must act sooner
And the real board confirms most of these structural signals.”
A Final Question
Finally, one more question — somewhat rhetorical from the perspective of Numerical Chess:
How could AI-Sha say so many correct things based solely on structural connections?
The answer is already contained in the Credo:
“Pieces create connections that generate measurable influence.”
Yes — pieces create connections.
Some of these connections can be used for subtractions (“capturing”) of opposing units.
Checkmate itself is the terminal connection.
The pieces themselves, numerically speaking, are variables.
Once a piece is subtracted, it can no longer create connections.
Connections are the true core of the game of chess.
That is precisely why a purely structural analysis of connections is already so informative.
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Numerical Chess
Exploring the hidden numerical structure of chess.

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Numerical Chess: Revealing the Numerical Structure of the Game
NUMERICAL CHESS – CURRENT STATE 2026
Episode 24 – Structural Analysis [8] – Recognizing and Counting Connections [7] – Exercise 6
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Numerical Chess — Credo
Chess positions are numerical structures.
Pieces create connections that generate measurable influence.
The game unfolds through the transformation of these numbers.
________________________________________
Exercise 6 – Recognizing All Structural Connections (IC, AC, BC)
In this final exercise, all structural connections for both sides must be counted in the attached position diagram.
The position is taken from the recent game between Jorden van Foreest and Magnus Carlsen at the TePe Sigeman & Co Tournament 2026.
Since the total number of connections in this position is quite large, a detailed list of individual connections has been omitted this time. Instead, the final numerical values for IC, AC, and BC in all zones can be found in the attached Numerical Chess Algorithm Table.
Correctly recognizing and counting all structural connections forms the foundation for every calculation in Numerical Chess.
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Numerical Chess
Exploring the hidden numerical structure of chess.

Numerical Chess: Revealing the Numerical Structure of the Game
NUMERICAL CHESS – CURRENT STATE 2026
Episode 23 – Structural Analysis [7] – Recognizing and Counting Connections [6] – Exercise 5
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Numerical Chess — Credo
Chess positions are numerical structures.
Pieces create connections that generate measurable influence.
The game unfolds through the transformation of these numbers.
________________________________________
Exercise 5 – Recognizing Base Connections (BC)

In this exercise, the task is to count all Base Connections (BC) for White and Black.
Base Connections represent the structural foundation from which AC and IC can develop.
The zonal BC values (shown in the blue rows of the table) are, as always, already entered in the Numerical Chess Algorithm Table.
Below you can find the individual Base Connections generated by White and Black in both strategic zones (20-zones) and their respective sub-zones.
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White
Zone 1
Ra1 → {b1,c1,d1,e1(x-ray),a2,a3}
Qc2 → {b1,c1,d1,a2,b2,d2,e2,c3}
Nc3 → {b1,d1,a2,e2}
Rd1 → {a1,b1,c1,e1,d2}
e2 → {d3,e3}
f2 → {e3}

BC = 26
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Zone 2
Ra1 → {d1,e1,f1,g1}
Qc2 → {d1,d2,e2}
Nc3 → {d1,e2}
Rd1 → {e1,f1,g1,d2}
e2 → {d3,e3,f3}
f2 → {e3,f3,g3}
Kg1 → {f1,h1,f2,g2,h2}
Bg2 → {f1,h1}
h2 → {g3,h3}

BC = 28

Black
Zone 1
Bb7 → {a8,c8,c6}
Qb6 → {d8,a7,b7,c7,c6}
Rc8 → {a8,b8,d8,e8,c7,c6}
Nd7 → {b8,b6}
Re7 → {e8,d7,e6}
f7 → {e6}

BC = 20
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Zone 2
Qb6 → {d8}
Rc8 → {d8,e8,f8,g8}
Nd7 → {f8}
Re7 → {e8,d7,f7,e6}
f7 → {e6,f6,g6}
Kg8 → {e8,h8,f7,g7,h7}
g7 → {f6,g6,h6}
h7 → {g6,h6}

BC =23
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Numerical Chess
Exploring the hidden numerical structure of chess.

Numerical Chess: Revealing the Numerical Structure of the Game
NUMERICAL CHESS – CURRENT STATE 2026
Episode 22 – Structural Analysis [6] – Recognizing and Counting Connections [5] – Exercise 4
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Numerical Chess — Credo
Chess positions are numerical structures.
Pieces create connections that generate measurable influence.
The game unfolds through the transformation of these numbers.
________________________________________
Exercise 4 – Recognizing Advancing Connections (AC)

In this exercise, the task is to count all Advancing Connections (AC) for White and Black.
The zonal AC values (shown in the green rows of the table) are, as always, already entered in the Numerical Chess Algorithm Table.
Below you can find the individual Advancing Connections generated by White and Black in both strategic zones (20-zones) and their respective sub-zones.
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White
Zone 1
a3 → {a4,b4}
b2 → {b4}
Nc3 → {a4,e4}
Rd1 → {d3,d4}
Nd4 → {b3}
Be2 → {d3,c4}
AC = 10
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Zone 2
Nc3 → {e4}
Rd1 → {d3,d4}
Nd4 → {f3}
Be2 → {d3,f3,g4}
g2 → {g4}
Rh1 → {h3,h4}
AC = 10

Black
Zone 1
Ra8 → {a6,a5}
b7 → {b5}
Nc5 → {a6}
Bd7 → {c6,b5}
Ne7 → {c6,d6}
e6 → {d5}
AC = 9
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Zone 2
Bd7 → {f5 – x-ray connection}
Ne7 → {d5,f5,g6}
e6 → {d5,f5}
f7 → {f5}
g7 → {g5}
AC = 8
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Numerical Chess
Exploring the hidden numerical structure of chess.

Numerical Chess: Revealing the Numerical Structure of the Game

NUMERICAL CHESS – CURRENT STATE 2026
Episode 21 – Structural Analysis [5] – Recognizing and Counting Connections [4] – Exercise 3
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Numerical Chess — Credo
Chess positions are numerical structures.
Pieces create connections that generate measurable influence.
The game unfolds through the transformation of these numbers.
________________________________________
Exercise 3 – Recognizing Advancing Connections (AC)
In this exercise, the task is to count all Advancing Connections (AC) for White.
The zonal AC values (shown in the green rows of the table) are, as always, already entered in the Numerical Chess Algorithm Table.
Below you can find the individual Advancing Connections generated by White in both strategic zones (20-zones) and their respective sub-zones.
The procedure for calculating connections in the subzones was explained in Episode 19.
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White
Zone 1
Qa1 → {a4 (battery connection), c3, d4}
Ra3 → {a4, b3, c3, d3, e3}
Nb5 → {c3, d4}
Re5 → {e4, e3}
Bf1 → {d3, c4}
AC = 14
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Zone 2
Qa1 → {d4}
Ra3 → {d3, e3, f3, g3, h3}
Nb5 → {d4}
Re5 → {e4, e3}
Bf1 → {d3}
g2 → {g4}
h2 → {h4}
AC = 12
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What are Advancing Connections?
Advancing Connections (AC) represent advancing pressure that may later develop into Invasive Connections (IC).
However, they also play an important role in defense, as they control forward lines and influence the opponent’s territory before actual invasion occurs.
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Question for readers
Which single white piece generates the largest number of Advancing Connections in this position?
Share your answer in the comments.
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Numerical Chess
Exploring the hidden numerical structure of chess.

"No chess player in the world can truly calculate whose thinking is not based on a numerical algorithm. The term 'calculation' is simply misused. True calculation is replaced here by visualization and other thought processes unrelated to true calculation." – NUMERICAL CHESS

Numerical Chess: Revealing the Numerical Structure of the Game

NUMERICAL CHESS – CURRENT STATE 2026

Episode 20 – Structural Analysis [4] – Recognizing and Counting Connections [3] – Exercise 2
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Numerical Chess — Credo
Chess positions are numerical structures.
Pieces create connections that generate measurable influence.
The game unfolds through the transformation of these numbers.
________________________________________
Exercise 2 – Recognizing Invasive Connections (IC)

In this exercise, the task is to count all Invasive Connections (IC) for both White and Black.
The final values are already entered in the Numerical Chess Algorithm Table.
If you attempted the exercise and would like to verify your results, the connections for both sides in the Strategic 20-Zones 1 and 2 are listed below.
(The sub-zone IC values can be calculated according to the rules explained in Episode 19.)
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White
20-Zone 1
Qa5→b5
Qa5→c5
Qa5→d5
Qa5→b6
Qa5→a6
Qa5→a7
Qa5→c7
Qa5→d8
Qa5→a8
Total: 9 IC
20-Zone 2
Qa5→d5
Qa5→d8
Total: 2 IC
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Black
20-Zone 1
Qb1→b4
d5→c4
d5→d4
d5→e4
Qb1→e4
Nf6→e4
Qb1→d3
Qb1→b3
Qb1→a2
Qb1→b2
Qb1→c2
Qb1→e1
Qb1→d1
Qb1→c1
Qb1→a1
Total: 15 IC
20-Zone 2
d5→d4
d5→e4
Qb1→e4
Nf6→e4
Nf6→g4
Qb1→d3
Qb1→d1
Qb1→e1
Qb1→f1
Qb1→g1
Qb1→h1
Total: 11 IC
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Numerical Chess
Exploring the hidden numerical structure of chess.