We demonstrate that, through appropriate prompting, GPT-3 family of models can be triggered to perform iterative behaviors necessary to execute (rather than just write or recall) programs that involve loops, including several popular algorithms found in computer science curricula or software developer interviews. We trigger execution and description of Iterations by Regimenting Self-Attention (IRSA) in one (or a combination) of three ways: 1) Using strong repetitive structure in an example of an execution path of a target program for one particular input, 2) Prompting with fragments of execution paths, and 3) Explicitly forbidding (skipping) self-attention to parts of the generated text. On a dynamic program execution, IRSA leads to larger accuracy gains than replacing the model with the much more powerful GPT-4. IRSA has promising applications in education, as the prompts and responses resemble student assignments in data structures and algorithms classes. Our findings hold implications for evaluating LLMs, which typically target in-context learning: We show that prompts that may not even cover one full task example can trigger algorithmic behavior, allowing solving problems previously thought of as hard for LLMs, such as logical puzzles. Consequently, prompt design is even more critical in LLM performance than previously recognized.