## 3 Results
### Analysis of the Weight of User Demand for Elderly Game Products  
Based on the extraction of user needs outlined in Table 3 and adhering to the principles of AHP model construction, a hierarchical analysis model was developed, as shown in Figure 9.  
After establishing the evaluation matrix, a focus group was invited to complete the assessment to ensure the scoring accuracy. Each level of need was compared and rated on a scale of 1 to 9. To construct a valid judgment matrix, AHP questionnaire responses were manually reviewed, and no significant outliers deviating from the mean were found. Therefore, the average method was applied to aggregate the scores for each comparison item, and the mean scores were calculated to derive the matrices in Equations (17)-(22). \[X= \begin{bmatrix}1&2&\frac{1}{2}&\frac{1}{2}&2\\ \frac{1}{2}&1&\frac{1}{3}&\frac{1}{5}&\frac{1}{2}\\ 2&3&1&\frac{1}{3}&3\\ 2&5&3&1&3\\ \frac{1}{2}&2&\frac{1}{3}&\frac{1}{3}&1\end{bmatrix}  \tag{17}\]  
\[X_{1}= \begin{bmatrix}1&5&2&3&5\\ \frac{1}{5}&1&\frac{1}{3}&\frac{1}{2}&1\\ \frac{1}{2}&3&1&\frac{1}{2}&5\\ \frac{1}{3}&2&2&1&5\\ \frac{1}{5}&1&\frac{1}{5}&1\end{bmatrix}  \tag{18}\]  
\[X_{2}= \begin{bmatrix}1&\frac{1}{2}&2\\ 2&1&3\\ \frac{1}{2}&\frac{1}{3}&1\end{bmatrix}  \tag{19}\]  
Figure 9: Hierarchical analysis model for elderly-friendly puzzle game product design. \[X_{3}= \begin{bmatrix}1&\frac{1}{3}&\frac{1}{5}&\frac{1}{2}&2&3\\ 3&1&\frac{1}{3}&2&3&5\\ 5&3&1&3&5&5\\ 2&\frac{1}{2}&\frac{1}{3}&1&2&3\\ \frac{1}{2}&\frac{1}{3}&\frac{1}{5}&\frac{1}{2}&1&1\\ \frac{1}{3}&\frac{1}{5}&\frac{1}{5}&\frac{1}{3}&1&1\end{bmatrix}  \tag{20}\]  
\[X_{4}= \begin{bmatrix}1&3&\frac{1}{3}&\frac{1}{3}\\ \frac{1}{3}&1&\frac{1}{2}&\frac{1}{3}\\ 2&2&1&\frac{1}{2}\\ 3&3&2&1\end{bmatrix}  \tag{21}\]  
\[X_{5}= \begin{bmatrix}1&2&\frac{1}{2}&\frac{1}{3}\\ \frac{1}{2}&1&\frac{1}{2}&\frac{1}{2}\\ 2&2&1&\frac{1}{2}\\ 3&2&2&1\end{bmatrix}  \tag{22}\]  
To validate the logical consistency of the judgment matrix and reliability of the results, weights were calculated using the geometric mean method. A consistency check was performed, and the results are listed in Table 5. All the \(CR\) values were below 0.1, confirming the feasibility of the calculations. To determine the comprehensive weight of each sub-criterion, this study applied a multiplicative calculation method that integrated the weights of each criterion layer with their corresponding sub-criterion weights. The resulting comprehensive weights are presented in Table 6.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Criterion Layer** & **Weight** & **Sub-Criterion Layer** & **Weight** & **Comprehensive Weight** & **Ranking** \\ \hline \multirow{4}{*}{X\({}_{1}\)} & & \(X_{11}\) & 0.419 & 0.0704 & 5 \\  & & \(X_{12}\) & 0.078 & 0.0131 & 20 \\  & 0.168 & \(X_{13}\) & 0.209 & 0.0351 & 11 \\  & & \(X_{14}\) & 0.233 & 0.0391 & 9 \\  & & \(X_{15}\) & 0.061 & 0.0102 & 22 \\ \hline \multirow{4}{*}{X\({}_{2}\)} & \multirow{4}{*}{0.072} & \(X_{21}\) & 0.297 & 0.0214 & 15 \\  & & \(X_{22}\) & 0.539 & 0.0388 & 10 \\  & & \(X_{23}\) & 0.164 & 0.0118 & 21 \\ \hline \multirow{4}{*}{X\({}_{3}\)} & \multirow{4}{*}{0.243} & \(X_{31}\) & 0.102 & 0.0248 & 14 \\  & & \(X_{32}\) & 0.225 & 0.0547 & 6 \\  & & \(X_{33}\) & 0.411 & 0.0999 & 3 \\ \cline{1-1}  & & \(X_{34}\) & 0.143 & 0.0347 &  
According to the AHP weight results, the primary themes identified in this study were simplicity and convenience (X\({}_{4}\), 0.407), leisure and entertainment (X\({}_{3}\), 0.243), and cognitive training methods and effects (X\({}_{1}\), 0.168), followed by comfort and safety (X\({}_{5}\), 0.110) and product appearance (X\({}_{2}\), 0.072). In the subsequent design phase, the two highest-ranking user needs for each theme were selected to guide product development.