All three invest the same monthly SIP into Nifty 50 over the same window. Best Day always buys at the month's lowest close (perfect timing). Worst Day always buys at the month's highest close (perfect mistiming). 15th just buys on the 15th (or next trading day). Spoiler: the gap is much smaller than you'd guess.
Why the gap stays narrow
Even perfect within-month timing, an oracle who always bought the cheapest close that month, only beats perfect mistiming by a few percent of XIRR. Why? A SIP buys ~360 times over 30 years. Within any one month, the high-low range is small relative to the long-run trend. The compounding from being invested swamps the day-picking edge.
The 15th-of-month investor, who picked a date once and never thought about it again, lands closer to the perfect-timing oracle than to the mistiming demon. That's the empirical case for just SIPping on a fixed date.