# Automatic activation of the shared-digit network in the solution of complex multiplication problems

## TL;DR

The study shows that complex multiplication problems automatically activate a network of single-digit problems sharing digits, affecting how quickly and accurately people solve them.

## Contribution

The study demonstrates that the shared-digit network is automatically activated during complex multiplication tasks, influencing performance.

## Key findings

- SDN size predicts speed and accuracy in solving complex multiplication problems.
- Participants solved problems with small SDNs more quickly than those with large SDNs.
- Retrieval was used more often for problems with small SDNs.

## Abstract

Previous studies have shown that the associative network of single-digit multiplication problems is automatically activated, even when participants perform irrelevant tasks, and that single-digit multiplication problems automatically activate all single-digit problems sharing at least one digit with them (the shared-digit network; SDN). We examined whether the SDN would also be automatically activated when participants perform an irrelevant task. Specifically, we asked whether complex multiplication problems (e.g., 2 × 12 = ) automatically activate all single-digit problems that share digits with them. In Experiment 1, participants solved all complex problems whose solutions were less than 100. In Experiments 2 and 3, participants solved sets of complex problems that differed in SDN size and in carryover status and were matched in problem size. Participants reported the strategies they used to solve the problems. Results showed that SDN size, which reflected the number of single-digit problems sharing digits with the complex problem, predicted speed and accuracy in the solution of complex problems. Regardless of carryover status and strategy, participants solved complex problems with small SDNs more quickly than complex problems with large SDNs. Regardless of carryover status, participants used retrieval more often when solving problems with a small SDN than with a large SDN. Thus, we have demonstrated that SDN size determines speed and accuracy in the solution of complex multiplication problems, and that the SDN is automatically activated even during performance of an irrelevant task.

The online version contains supplementary material available at 10.3758/s13421-025-01766-1.