Tool use, such as web search, has become a standard capability even in freely available large language models (LLMs). However, existing benchmarks evaluate temporal reasoning mainly in static, non-tool-using settings, which poorly reflect how LLMs perform temporal reasoning in practice. We introduce Time Puzzles, a constraint-based date inference task for evaluating iterative temporal reasoning with tools. Each puzzle combines factual temporal anchors with (cross-cultural) calendar relations and may admit one or multiple valid dates. The puzzles are algorithmically generated, enabling controlled and continual evaluation. Across 13 LLMs, even the best model (GPT-5) achieves only 55.3% accuracy without tools, despite using easily searchable facts. While web search improves performance, models perform substantially better when constraints are rewritten with explicit dates, removing the need for factual lookup. These results reveal a gap in reliable tool use for iterative temporal reasoning.