High-z Sudoku: A diagnostic tool for identifying robust (sub)mm redshifts

We present methods to (i) graphically identify robust redshifts using emission lines in the (sub)mm regime, (ii) evaluate the capabilities of different (sub)mm practices for measuring spectroscopic redshifts, and (iii) optimise future (sub)mm observations towards increasing the fraction of robust redshifts. Using this publicly-available code (https://github.com/tjlcbakx/redshift-search-graphs), we discuss scenarios where robust redshifts can be identified using both single- and multiple-line detections, as well as scenarios where the redshift remains ambiguous, even after the detection of multiple lines. Using the redshift distribution of (sub)mm samples, we quantify the efficiencies of various practices for measuring spectroscopic redshifts, including interferometers, as well as existing and future instruments specifically designed for redshift searches. Finally, we provide a method to optimise the observation strategy for future (sub)mm spectroscopic redshift searches with the Atacama Large Millimetre/submillimetre Array, where 2 mm proves indispensable for robust redshifts in the z = 2 - 4 region.