A digital clock shows the time as HH:MM on a 24-hour dial. Turned 180 degrees it must still read as the same time. The turn brings the minutes to the front and the hours to the back, so the minutes must be the hours reversed. Find every such time from 03:30 to 21:00.
Your answer
Watch outReversing the hour looks sufficient, but 06 would need minute 60. The minutes' tens digit must be 5 or less.
How to solve
StrategyMake a Systematic List — Flipping the display 180 degrees turns HH:MM into (reverse of MM):(reverse of HH). For nothing to change, the minutes have to read as the hours reversed - the time has the shape ab:ba. So I systematically list every ab:ba time inside the range, checking the hour and minute are valid. Spotting the ab:ba shape is the pattern that makes the list short.
1STEP 1
See what a 180-degree turn does to the display
A 180-degree turn reverses the order: minutes first, hours last.
Mentally rotating the clock face is the heart of the puzzle; reading a digital clock is a Grade 2 skill.
2.MD.C.7Visualize Spatial Relationships
2STEP 2
Turn the picture into a digit rule
So hour ab needs minute ba — the shape is ab:ba.
HH:MM = ab : ba
Seeing a two-digit hour as a tens digit and a ones digit, then reversing them, uses Grade 2 place-value thinking.
2.NBT.A.1Look For A Pattern
Turning the display through half a turn swaps the tens digit and the ones digit of the hour.
Why?
A half turn moves every digit to the opposite end of the display without stretching or bending any of them.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
A two-digit hour is a tens count and a ones count, so exchanging which digit sits in which place makes a different hour.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
3STEP 3
List the ab:ba times from 03:30 to 21:00
Dropping 06-09 and 16-19, whose minutes overflow, leaves 10 times.
Separating given examples from new answers makes the final list clear.
2.MD.C.7Make A Systematic List
Answer
03:30, 04:40, 05:50, 10:01, 11:11, 12:21, 13:31, 14:41, 15:51, 20:02 (10 times; 8 of them new to write)
Every listed time has the form ab:ba, so flipping it returns the same reading - 11:11 and 12:21 are easy to verify by eye. Each minute value (30,40,50,01,11,21,31,41,51,02) is a real minute under 60, and each hour (03-20) is inside the 03:30-21:00 window, so all ten are valid and none outside the range was missed.
Takeaway
A flipped time reads the same only when the minutes are the hours written backward - spotting that ab:ba pattern needs just Grade 2 place value and clock reading!