A feedback-based word deduction framework based on the Jotto problem is proposed, and the problem space is represented as a weighted graph where all valid words correspond to nodes, and the edge weight is defined by the number of common letters between the two words. Finally, the gameplay is defined as an iterative constraint propagation mechanism where feedback is used to iteratively narrow the incompatible space of the graph, facilitating the reduction of the hypothesis space in a structured and interpretable manner. In contrast to existing approaches, where the problem space is typically defined for fixed-length isograms, the proposed framework generalizes to variable-length words (between 3 and 8 letters) and naturally extends to repeated letter cases, facilitating the treatment of realistic Jotto problem instances within a unified framework for the first time. The proposed framework's applicability and solver dynamics are also discussed through an interactive implementation and a qualitative case study, respectively. Significant automated tests on approximately 3,000 simulated gameplay scenarios identify a novel convergence behavior: the expected number of iterations diminishes with increasing word length. A strong relationship is confirmed using statistical tests to verify a logarithmic relationship, which is also verified using regression modeling and goodness-of-fit tests. In addition to the initial problem statement, this formulation introduces graph pruning as a viable paradigm for feedback-driven inference with interpretability and its association with symbolic reasoning and interactive intelligent systems.
Figures & tables
Word Length
All Words
Unique Letter Words
3
1295
1176
4
4995
3836
5
9972
6373
6
17464
8077
7
15832
7882
8
23768
6074
Table 1: Dictionary size based on word length and character uniqueness
Figure 1: Schematic overview of the proposed graph-based inference workflow.
Figure 2: Average number of iterations required to identify the target word as a function of word length. A clear decreasing trend is observed, indicating faster convergence for longer words.
Model
Regression Equation
Linear
y=−1.014x+13.995
Exponential
y=16.170e−0.123x
Logarithmic
y=−5.253ln(x)+17.094
Quadratic
y=0.089x2−1.996x+16.436
Table 2: Analytical expressions of the fitted regression models (parameters rounded to three decimal places).
Model
R2
SSE
χ2
AIC
Linear
0.962
0.705
0.072
8.177
Exponential
0.978
0.411
0.044
4.941
Logarithmic
0.984
0.295
0.035
2.946
Quadratic
0.978
0.407
0.047
6.885
Table 3: Comparison of regression models using goodness-of-fit metrics. The logarithmic model provides the best overall fit across all evaluation criteria.
Figure 3: Comparison of fitted regression models showing the relationship between word length and average iteration count. The logarithmic model provides the closest agreement with empirical observations.
Figure 4: Representative lexical overlap network with approximately ∼700 nodes. Nodes correspond to valid four-letter English words, while edges encode shared-letter relationships. Only edges with at least two shared letters are shown for visual clarity.
Figure 5: Progressive contraction of the candidate network during the final stages of convergence. Only the last five iterations are shown for clarity, illustrating how feedback-driven pruning isolates the target word good .
Iteration
Guess
Feedback a
Graph Nodes
1
fain
0
1549
2
roue
1
860
3
lots
1
393
4
hyle
0
98
5
comb
1
41
6
scud
1
13
Table 4: Iteration-by-iteration suggestions and feedback for the secret word “good”.
Component
Best Case
Expected Case
Worst Case
Feedback evaluation
O(L)
O(L)
O(L)
Candidate filtering
O(NL)
O(NL)
O(NL)
Heuristic selection
O(1)
O(NL)
O(N2L)
Graph construction
O(N)
O(NkL)
O(N2L)
Total runtime
O(N0L)
O(N0LlogN0)
O(N02L)
Space complexity
O(N)
O(N+E)
O(N2)
Table 5: Asymptotic complexity profile of the proposed solver. Expected-case bounds reflect adaptive pruning dynamics observed during empirical evaluation.