Solution to Enigma No. 1574

Answer: 12 April 2009 and 4 December 2009

The winner is Geoff Stone of North Bayswater, Victoria, Australia. There were 140 entries.

Read the original question

Worked answer

A date that starts with the month must have six digits, the first two of which lie between 01 and 12, and cannot be a square of a number greater than 350.

In the lists below, if one regards each square as a 6-digit number abcdef, squares marked A are ineligible because cd is a number greater than the number of days in the month ab, and those marked B are ineligible because the number cdabef is not a perfect square.

Squares that end in 01 are squares of numbers that end in 01, 49, 51 or 99.

• 201² = 040401 (given)
• 249² = 062001 B
• 251² = 063001 B
• 299² = 089401 A
• 301² = 090601 B
• 349² = 121801 B

No squares end in 2 or 3. Squares that in 04 are the squares of numbers that end in 02, 48, 52, 98.

• 102² = 010404 B
• 148² = 021904 B
• 152² = 023104 A
• 198² = 039204 A
• 202² = 040804 B
• 248² = 061504 B
• 252² = 063504 A
• 298² = 088804 A
• 302² = 091204 B
• 348² = 121104 B

Squares that end in 5 all in 25; those that end in 6 all have penultimate digit odd. No squares end in 7 or 8. Those that end in 09 are squares of numbers that end in 03, 47, 53 or 97.

• 103² = 010609 B
• 147² = 021609 B
• 153² = 023409 A
• 197² = 038809 A
• 203² = 041209
• 247² = 061009 B
• 253² = 064009 A
• 297² = 088209 A
• 303² = 091809 B
• 347² = 120409

So 041209 and 120409 are both squares, making 12 April 2009 and 4 December 2009 both double square dates.