by Brian Tomasik
Completed in 2002

## Background

The present document contains potential answers to a “clock problem” invented by one of my former teachers. In the beginning of 2002, I and the other members of my high-school Math Club began work on the listed solutions. I finished the rest myself over winter vacation that year.

## Rules

1. For each time shown on a digital clock, use all of the digits in an equation with only one equals sign.
2. No extra numbers can be added to the equation, only mathematical symbols or operations (such as +, -, ∙, ٪, √, !, sin, cos, tan, sec, csc, cot, and parentheses).
3. Exponents or roots (other than a square root) may only be used if the number written in the symbol appears in the correct order in the time. For example, 3√ may only be used in a time such as 2:38 (2 = 3√8).
4. All numbers in the equation must be in their order on the clock (so, for example, if the time is 2:42, you may not switch the second 2 and 4 to create the following: 22 = 4).
5. The symbol π may not be used (because, if it were allowed, any integer could be created by adding or subtracting π/π one or several times).
6. Only an equals sign may be inserted, not ≠, ≈, ≤, ≥, <, or >.

## Digit conversions

Trigonometric functions allow for converting a single digit into a different digit. Without this trick, some of the clock times would be extremely difficult to solve.

Here are the digit substitutions:

• (2→1) is short for tan(arc sec(√(2))), which equals 1
• (√(4)→1) is short for tan(arc sec(√(√(4)))), which equals 1
• (3→.5) is short for cos(arc tan(√(3))), which equals .5
• (.5→1) is short for tan(arc sec(√(sec(arc cos(.5))))), which equals 1
• (3→1) is short for tan(arc sec(√(sec(arc tan(√(3)))))), which equals 1
• (1→0) is short for sin(arc cos(1)), which equals 0
• (2→0) is short for sin(arc cos(tan(arc sec(√(2))))), which equals 0
• (√(4)→0) is short for sin(arc cos(tan(arc sec(√(√(4)))))), which equals 0
• (.5→0) is short for sin(arc cos(tan(arc sec(√(sec(arc cos(.5))))))), which equals 0
• (3→0) is short for sin(arc cos(tan(arc sec(√(sec(arc tan(√(3)))))))), which equals 0
• (1→√(2)) is short for sec(arc tan(1)), which equals √(2)
• (0→√(2)) is short for sec(arc tan(cos(0))), which equals √(2)

## Solutions

1:00    1∙0 = 0
1:01    1+0 = 1
1:02    1+0! = 2
1:03    1 = (0!)3
1:04    1+0! = √(4)
1:05    1 = (0!)5
1:06    1 = (0!)6
1:07    1 = (0!)7
1:08    1 = (0!)8
1:09    1 = (0!)9
1:10    1-1 = 0
1:11    1∙1 = 1
1:12    1+1 = 2
1:13    1 = 13
1:14    1 = 14
1:15    1 = 15
1:16    1 = 16
1:17    1 = 17
1:18    1 = 18
1:19    1 = 19
1:20    1 = 20
1:21    1 = 2-1
1:22    1 = 2/2
1:23    -1 = 2-3
1:24    1∙2 = √(4)
1:25    1 = 2∙.5
1:26    (1+2)! = 6
1:27    1 = (2→1)7
1:28    1 = (2→1)8
1:29    1+2 = √(9)
1:30    1 = 30
1:31    13 = 1
1:32    1 = 3-2
1:33    1∙3 = 3
1:34    1+3 = 4
1:35    1-3! = -5
1:36    1∙3! = 6
1:37    1+3! = 7
1:38    1 = (.(-3+8)→1)
1:39    1∙3 = √9
1:40    1 = 40
1:41    1 = √(4)-1
1:42    1∙√(4) = 2
1:43    1+√(4) = 3
1:44    1∙4 = 4
1:45    1+4 = 5
1:46    (-1+4)! = 6
1:47    1 = (√(4)→1)7
1:48    1 = (√(4)→1)8
1:49    1+√(4) = √(9)
1:50    15 = 0!
1:51    15 = 1
1:52    1 = .5∙2
1:53    1+5 = 3!
1:54    1 = 5-4
1:55    1 = 5/5
1:56    1+5 = 6
1:57    1 = [(-5+7)→1]
1:58    1 = (√(.5∙8)→1)
1:59    1+5 = (√(9))!
2:00    2∙0 = 0
2:01    20 = 1
2:02    2+0 = 2
2:03    2+0! = 3
2:04    2 = 0+√(4)
2:05    2 = √(-0!+5)
2:06    (2+0!)! = 6
2:07    (2→1) = (0!)7
2:08    (2→1) = (0!)8
2:09    2+0! = √(9)
2:10    2 = 1+0!
2:11    2 = 1+1
2:12    2 = 1∙2
2:13    2+1 = 3
2:14    2∙1 = √(4)
2:15    2 = √(-1+5)
2:16    (2+1)! = 6
2:17    (2→1) = 17
2:18    (2→1) = 18
2:19    2+1 = √(9)
2:20    2 = 2+0
2:21    2 = 2∙1
2:22    2 = √(2) ∙√(2)
2:23    2 = √(-2+3!)
2:24    2+2 = 4
2:25    (2→1) = 2∙.5
2:26    2 = √(-2+6)
2:27    (2→1) = (2→1)7
2:28    2∙√(2) = √(8)
2:29    (2→1)+2 = √(9)
2:30    2 = 3-0!
2:31    2 = 3-1
2:32    2 = 3-(2→1)
2:33    2 = (3!)/3
2:34    2 = 3!-4
2:35    2+3 = 5
2:36    2∙3 = 6
2:37    2 = √(-3+7)
2:38    2 = 3√(8)
2:39    2 = (3!)/(√(9))
2:40    2 = √(4+0)
2:41    2 = √(4∙1)
2:42    2 = 4-2
2:43    2+4 = 3!
2:44    2 = 4/(√(4))
2:45    2 = .4∙5
2:46    2+4 = 6
2:47    (2→1) = (√(4)→1)7
2:48    2∙4 = 8
2:49    2 = -4+(√(9))!
2:50    2 = √(5-0!)
2:51    2 = √(5-1)
2:52    (2→1) = .5∙2
2:53    2 = 5-3
2:54    2 = .5∙4
2:55    √(25) = 5
2:56    (-2+5)! = 6
2:57    2+5 = 7
2:58    2 = √(.5∙8)
2:59    2 = 5-√(9)
3:00    3∙0 = 0
3:01    (3∙0)! = 1
3:02    3-0! = 2
3:03    3-0 = 3
3:04    3 = -0!+4
3:05    sin(30°) = .5
3:06    3!+0 = 6
3:07    3!+0! = 7
3:08    3 = √(0!+8)
3:09    3+0 = √(9)
3:10    [(3-1)→1] = 0!
3:11    [(3-1)→1] = 1
3:12    3 = 1+2
3:13    3∙1 = 3
3:14    3+1 = 4
3:15    3! = 1+5
3:16    3!∙1 = 6
3:17    3!+1 = 7
3:18    3 = √(1+8)
3:19    3∙1 = √(9)
3:20    3 = 2+0!
3:21    3 = 2+1
3:22    3-2 = (2→1)
3:23    3! = 2∙3
3:24    3-2 = (√(4)→1)
3:25    3+2 = 5
3:26    3∙2 = 6
3:27    3 = √(2+7)
3:28    3!+2 = 8
3:29    32 = 9
3:30    3 = 3+0
3:31    3 = 31
3:32    3 = √(32)
3:33    3! = 3+3
3:34    3! = 3∙√(4)
3:35    3 = 3!∙.5
3:36    3+3 = 6
3:37    (3→1) = (3→1)7
3:38    (3→.5) = .(-3+8)
3:39    3∙3 = 9
3:40    3 = 4-0!
3:41    3 = 4-1
3:42    3! = 4+2
3:43    3! = √(4)∙3
3:44    3! = 4+√(4)
3:45    -3 = √(4)-5
3:46    3∙√(4) = 6
3:47    3+4 = 7
3:48    3!+√(4) = 8
3:49    3! = √(4)∙ √(9)
3:50    3! = 5+0!
3:51    3! = 5+1
3:52    3 = 5-2
3:53    3!∙.5 = 3
3:54    3 = 5-√(4)
3:55    (3→.5) = √(.5)∙ √(.5)
3:56    3 = .5∙6
3:57    (3→1) = (.5→1)7
3:58    3+5 = 8
3:59    3!∙.5 = √(9)
4:00 to 4:59 are the same as 2:00 to 2:59, as √(4) = 2
5:00    5∙0 = 0
5:01    50 = 1
5:02    .5 = (0!)/2
5:03    5+0! = 3!
5:04    5 = 0!+4
5:05    5+0 = 5
5:06    5+0! = 6
5:07    (.5→1) = (0!)7
5:08    .5 = (√(0!+8)→.5)
5:09    5+0! = (√(9))!
5:10    √(√(5-1)) = sec(arc tan(0!))
5:11    √(√(5-1)) = sec(arc tan(1))
5:12    .5 = 1/2
5:13    -5 = 1-3!
5:14    5-1 = 4
5:15    5∙1 = 5
5:16    5+1 = 6
5:17    (.5→1) = 17
5:18    .5 = (√(1+8)→.5)
5:19    -5 = 1-(√(9))!
5:20    .5 = [(2+0!)→.5]
5:21    .5 = [(2+1)→.5]
5:22    5∙.2 = (2→1)
5:23    5 = 2+3
5:24    5 = 2/(.4)
5:25    5 = √(25)
5:26    (5-2)! = 6
5:27    5+2 = 7
5:28    (.5→1) = (2→1)8
5:29    5 = 2+√(9)
5:30    .5 = sin(30°)
5:31    5 = 3!-1
5:32    5 = 3+2
5:33    .5 = 3/(3!)
5:34    5 = 3+√(4)
5:35    (.5→1) = (3→1)∙(.5→1)
5:36    .5 = 3/6
5:37    (.5→1) = (3→1)7
5:38    5+3 = 8
5:39    .5 = 3/[(√9)!]
5:40    5 = 4+0!
5:41    5 = 4+1
5:42    5∙.4 = 2
5:43    5 = √(4)+3
5:44    5∙.4 = √(4)
5:45    .5 = (√(4+5)→.5)
5:46    [(.5→1)+√(4)]! = 6
5:47    (.5→1) = (√(4)→1)7
5:48    (.5→1) = (√(4)→1)8
5:49    5+4 = 9
5:50    5 = 5+0
5:51    5 = 5∙1
5:52    (.5→1) = 5∙.2
5:53    (.5→1) = (.5→1)3
5:54    (.5→1) = (.5→1)4
5:55    5 = √(5)∙√(5)
5:56    (.5→1) = (.5→1)6
5:57    (.5→1) = (.5→1)7
5:58    (.5→1) = (.5→1)8
5:59    (.5→1) = (.5→1)9
6:00    6∙0 = 0
6:01    (6∙0)! = 1
6:02    6 = (0!+2)!
6:03    6 = 0+3!
6:04    6 = (-0!+4)!
6:05    6 = 0!+5
6:06    6 = 0+6
6:07    6 = -0!+7
6:08    6 = (√(0!+8))!
6:10    6∙(1→0) = 0
6:11    (6∙(1→0))! = 1
6:12    6 = (1+2)!
6:13    6 = (1∙3)!
6:14    6 = (-1+4)!
6:15    6 = 1+5
6:16    6 = 1∙6
6:17    6 = -1+7
6:18    6 = (√(1+8))!
6:19    6 = (√(1∙9))!
6:20    6 = (2+0!)!
6:21    6 = (2+1)!
6:22    6 = (2+(2→1))!
6:23    6 = 2∙3
6:24    6 = 2+4
6:25    6 = (2→1)+5
6:26    6 = (2→1)∙6
6:27    6 = -(2→1)+7
6:28    6 = -2+8
6:29    6 = (√[(2→1)∙9])!
6:30    6 = (3+0)!
6:31    6 = (3∙1)!
6:32    6 = 3∙2
6:33    6 = (√(3∙3))!
6:34    6 = 3∙√(4)
6:35    6 = (3→1)+5
6:36    6 = (-3+6)!
6:37    -6 = (3→1)-7
6:38    6 = (√[(3→1)+8])!
6:39    6 = -3+9
6:40    6 = (√(4)+0!)!
6:41    6 = (√(4)+1)!
6:42    6 = 4+2
6:43    6 = √(4)∙3
6:44    6 = 4+√(4)
6:45    6 = (√(4)→1)+5
6:46    6∙(√(4)→1) = 6
6:47    6+(√(4)→1) = 7
6:48    6+√(4) = 8
6:49    6 = √(4)∙√(9)
6:50    6 = 5+0!
6:51    6 = 5+1
6:52    6 = (5-2)!
6:53    6∙.5 = 3
6:54    6 = 5+(√(4)→1)
6:55    6 = 5+(.5→1)
6:56    6 = (.5→0)+6
6:57    6 = -(.5→1)+7
6:58    6 = (-5+8)!
6:59    6∙.5 = √(9)
7:00    7∙0 = 0
7:01    (7∙0)! = 1
7:02    7∙0 = (2→0)
7:03    7∙0 = (3→0)
7:04    7∙0 = (√(4)→0)
7:05    7∙0 = (.5→0)
7:06    7 = 0!+6
7:07    7+0 = 7
7:08    7+0! = 8
7:09    7-0! = (√(9))!
7:10    7∙(1→0) = 0
7:11    (7∙(1→0))! = 1
7:12    (7∙(1→0))! = (2→1)
7:13    7-1 = 3!
7:14    (7∙(1→0))! = (√(4)→1)
7:15    (7∙(1→0))! = (.5→1)
7:16    7 = 1+6
7:17    7∙1 = 7
7:18    7+1 = 8
7:19    7-1 = (√(9))!
7:20    7∙(2→0) = 0
7:21    (7∙(2→0))! = 1
7:22    (7∙(2→0))! = (2→1)
7:23    7-(2→1) = 3!
7:24    (7∙(2→0))! = (√(4)→1)
7:25    7 = 2+5
7:26    7-(2→1) = 6
7:27    7∙(2→1) = 7
7:28    7+(2→1) = 8
7:29    7∙(2→0) = (√(9)→0)
7:30    7 = 3!+0!
7:31    7 = 3!+1
7:32    7 = 3!+(2→1)
7:33    7 = 3!+(3→1)
7:34    7 = 3+4
7:35    7 = 3!+(.5→1)
7:36    7 = (3→1)+6
7:37    7∙(3→1) = 7
7:38    7+(3→1) = 8
7:39    7∙(3→0) = (√(9)→0)
7:40    7∙(√(4)→0) = 0
7:41    (7∙(√(4)→0))! = 1
7:42    (7∙(√(4)→0))! = (2→1)
7:43    7 = 4+3
7:44    (7∙(√(4)→0))! = (√(4)→1)
7:45    7-√(4) = 5
7:46    7 = (√(4)→1)+6
7:47    7∙(√(4)→1) = 7
7:48    7+(√(4)→1) = 8
7:49    7-4 = √(9)
7:50    7∙(.5→0) = 0
7:51    (7∙(.5→0))! = 1
7:52    7 = 5+2
7:53    7∙(.5→0) = (3→0)
7:54    7 = 5+√(4)
7:55    7∙(.5→0) = (.5→0)
7:56    7 = (.5→1)+6
7:57    7 = (.5→1)∙7
7:58    7+(.5→1) = 8
7:59    [(7-5)→1] = (√(9)→1)
8:00    8∙0 = 0
8:01    (8∙0)! = 1
8:02    (8∙0)! = (2→1)
8:03    √(8+0!) = 3
8:04    √(8) = (0→√(2))∙√(4)
8:05    8∙0 = (.5→0)
8:06    (√(8+0!))! = 6
8:07    8 = 0!+7
8:08    8+0 = 8
8:09    8+0! = 9
8:10    8∙(1→0) = 0
8:11    (8∙(1→0))! = 1
8:12    √(8) = (1→√(2))∙2
8:13    8∙(1→0) = (3→0)
8:14    √(8) = (1→√(2))∙√(4)
8:15    8∙(1→0) = (.5→0)
8:16    (√(8+1))! = 6
8:17    8 = 1+7
8:18    8∙1 = 8
8:19    8+1 = 9
8:20    8∙(2→0) = 0
8:21    8∙(2→0) = (1→0)
8:22    √(8) = 2∙√(2)
8:23    8 = 2+3!
8:24    8 = 2∙4
8:25    8∙(2→0) = (.5→0)
8:26    8 = 2+6
8:27    8 = (2→1)+7
8:28    8 = (2→0)+8
8:29    (√(8/2)→1) = (√(9)→1)
8:30    8∙(3→0) = 0
8:31    [(8-3!)→1] = 1
8:32    8 = 3!+2
8:33    8∙(3→0) = (3→0)
8:34    8 = 3!+√(4)
8:35    8 = 3+5
8:36    (√(8+(3→1)))! = 6
8:37    8 = (3→1)+7
8:38    8∙(3→1) = 8
8:39    8+(3→1) = 9
8:40    8∙(√(4)→0) = 0
8:41    [(8/4)→1] = 1
8:42    8/4 = 2
8:43    8∙(√(4)→0) = (3→0)
8:44    8 = 4+4
8:45    8 = 4/(.5)
8:46    8 = √(4)+6
8:47    8 = (√(4)→1)+7
8:48    8∙(√(4)→1) = 8
8:49    8+(√(4)→1) = 9
8:50    8∙(.5→0) = 0
8:51    8∙(.5→0) = (1→0)
8:52    √(8∙.5) = 2
8:53    8 = 5+3
8:54    8∙.5 = 4
8:55    (√(8∙.5)→1) = (.5→1)
8:56    (8-5)! = 6
8:57    8 = (.5→1)+7
8:58    8∙(.5→1) = 8
8:59    8 = 5+√(9)
9:00 to 9:59 are the same as 3:00 to 3:59, as √(9) = 3
10:00    10 = 00
10:01    1+0 = 0+1
10:02    1+0! = 0+2
10:03    1+0!+0! = 3
10:04    1+0!+0 = √(4)
10:05    1∙0 = 0∙5
10:06    (1+0!+0!)! = 6
10:07    1∙0 = 0∙7
10:08    1 = (00)8
10:09    1+0!+0! = √(9)
10:10    10 = 10
10:11    1+0! = 1+1
10:12    1+0+1 = 2
10:13    1+0!+1 = 3
10:14    1+0+1 = √(4)
10:15    1+0! = 1/(.5)
10:16    (1+0!+1)! = 6
10:17    1+0 = 17
10:18    1+0 = 18
10:19    1+0 = 19
10:20    -1+0+2 = 1
10:21    1+0 = 2-1
10:22    1+0 = 2/2
10:23    1+0+2 = 3
10:24    1+0!+2 = 4
10:25    1+0 = .2∙5
10:26    (1+0+2)! = 6
10:27    1+(0!+2)! = 7
10:28    (1+0!)∙√(2) = √(8)
10:29    1+0+2 = √(9)
10:30    1+0! = 3-0!
10:31    1+0! = 3-1
10:32    1+0 = 3-2
10:33    1+0 = 3/3
10:34    10 = 3!+4
10:35    1+0!+3 = 5
10:36    (1+0!)∙3 = 6
10:37    1+0+3! = 7
10:38    1+0!+3! = 8
10:39    1-0!+3 = √(9)
10:40    1+0! = √(4)+0
10:41    1+0! = √(4)∙1
10:42    1+0! = 4/2
10:43    1+0 = 4-3
10:44    1+0 = 4/4
10:45    1+0! = .4∙5
10:46    1+0! = -4+6
10:47    1+(0!+√(4))! = 7
10:48    (1+0!)∙4 = 8
10:49    -1+0+4 = √(9)
10:50    10 = 50
10:51    1+0/5 = 1
10:52    1+0 = .5∙2
10:53    1 = -0!+5-3
10:54    1+0 = 5-4
10:55    1+0 = 5/5
10:56    1+0+5 = 6
10:57    1+0!+5 = 7
10:58    1+0! = √(.5∙8)
10:59    1+0! = √(-5+9)
11:00    1-1 = 0-0
11:01    1+1 = 0!+1
11:02    1+1+0 = 2
11:03    1+1+0! = 3
11:04    1+1+0 = √(4)
11:05    1+1 = √(-0!+5)
11:06    (1+1+0!)! = 6
11:07    1-1 = 07
11:08    1-1 = 08
11:09    1+1+0! = √(9)
11:10    1+1 = 1+0!
11:11    1+1 = 1+1
11:12    1+1 = 1∙2
11:13    1+1+1 = 3
11:14    1+1∙1 = √(4)
11:15    1 = 115
11:16    (1+1+1)! = 6
11:17    1 = 117
11:18    1∙1 = 18
11:19    1+1+1 = √(9)
11:20    1+1 = 2+0
11:21    1+1 = 2∙1
11:22    1/1 = 2/2
11:23    -1/1 = 2-3
11:24    (1+1)∙2 = 4
11:25    1/1 = .2∙5
11:26    1 = [(1+2)!]/6
11:27    1+(1+2)! = 7
11:28    √(1+1)∙2 = √(8)
11:29    1/1+2 = √(9)
11:30    1+1 = 3-0!
11:31    1+1 = 3-1
11:32    1/1 = 3-2
11:33    1/1 = 3/3
11:34    1/1+3 = 4
11:35    1+1+3 = 5
11:36    (1+1)∙3 = 6
11:37    1/1+3! = 7
11:38    1+1+3! = 8
11:39    1/1∙3 = √(9)
11:40    1-1 = 4∙0
11:41    1 = √(1∙4)-1
11:42    1+1 = 4/2
11:43    1/1 = 4-3
11:44    1-1 = 4-4
11:45    1+1 = 4∙.5
11:46    1+1+4 = 6
11:47    1 = 147
11:48    (1+1)∙4 = 8
11:49    1/1+√(4) = √(9)
11:50    (1-1)5 = 0
11:51    (1+1)∙.5 = 1
11:52    1∙1/(.5) = 2
11:53    1+1 = 5-3
11:54    1/1 = 5-4
11:55    1∙1 = 5/5
11:56    1/1+5 = 6
11:57    1+1+5 = 7
11:58    1+1 = √(.5∙8)
11:59    1 = 159
12:00    1/2∙0 = 0
12:01    -1+2+0 = 1
12:02    1∙2 = 0+2
12:03    1+2+0 = 3
12:04    1+2+0! = 4
12:05    (1+2)!-0! = 5
12:06    (1+2)!+0 = 6
12:07    (1+2)!+0! = 7
12:08    1∙(2→0) = 0∙8
12:09    1+2+0 = √(9)
12:10    1-2+1 = 0
12:11    1-2/1 = -1
12:12    1+2 = 1+2
12:13    1+2 = 1∙3
12:14    1+2+1 = 4
12:15    1∙2 = 1/(.5)
12:16    1∙2 = √(√(16))
12:17    1+(2+1)! = 7
12:18    1∙(2→1) = 18
12:19    1+2∙1 = √(9)
12:20    1 = 2/2+0
12:21    1+2 = 2+1
12:22    1∙2 = √(2∙2)
12:23    (1+2)! = 2∙3
12:24    1∙2+2 = 4
12:25    1+2+2 = 5
12:26    (-1+2+2)! = 6
12:27    1 = 2∙.(-2+7)
12:28    1∙2∙√(2) = √(8)
12:29    -1+2+2 = √(9)
12:30    1+2 = 3+0
12:31    1∙2 = 3-1
12:32    (1+2)! = 3∙2
12:33    1+2 = √(3)∙√(3)
12:34    1-2 = 3-4
12:35    1∙2+3 = 5
12:36    1∙2∙3 = 6
12:37    1+2∙3 = 7
12:38    1∙2+3! = 8
12:39    (1+2)∙3 = 9
12:40    1∙2 = √(4+0)
12:41    √(√(√(12+4))) = (1→√(2))
12:42    1∙2 = 4/2
12:43    (1+2)! = √(4)∙3
12:44    1∙2∙√(4) = 4
12:45    1-2 = 4-5
12:46    1∙2+4 = 6
12:47    1+2+4 = 7
12:48    1/2 = 4/8
12:49    1+2∙4 = 9
12:50    1 = 2∙.5+0
12:51    1∙2∙.5 = 1
12:52    1+2 = 5-2
12:53    -1 = √(25)-3!
12:54    -1+2 = 5-4
12:55    1∙√(25) = 5
12:56    (1+2)/(.5) = 6
12:57    1∙2 = -5+7
12:58    1∙2 = √(.5∙8)
12:59    (1+2)!∙.5 = √(9)