The last puzzle is constructed of the first 24 solutions.

Solution below.

Reveal solution by highlighting:

Note that there are 3 meta-clues, talking about other clues.

Clue 9 tells me, that clue 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20 and 24 are false. I sort of suspected something like that, because clue 1-8 appeared to be enough to find a solution, but testing said it wasn’t. None of the solutions I tried worked.

Clue 23 tells me, clue 23 is false. There is therefore no way clue 23 can be true. To put everything right, the first part of the clue, that clue 9 is false, must be false. “Clue 9 is false” = false. “Clue 23 is false” = true. And false and true = false. Therefore clue 9 is true (and clue 23 false). Whew! Clue 9 is true, that’s a relief. Now we know those clues, fingered by clue 9, are in fact all false.

Clue 24 tells me, there are less than 10 true clues. But clue 9 tells me, clue 24 is false. Therefore there are 10 or more true clues. Clue 9 tells me there are 12 false clues. Clue 23 added 1 more false clue. 13 so far. That would mean 11 true clues as well. So there might be 1 more false clue lurking out there.

Clue 16 and clue 17 can’t both be true. So we have 1 candidate right there.

Let’s look at what we have so far. We’re looking for a 5 digit number, ABCDE.

Day	Clue	True?
1	C is 1, 8, or 0.	False
2	C is 6, 8, or 1.	False
3	A is 5.	False
4	E is 2 or 6.	False
5	D is 3, 7, or 8.	False
6	E is 2, 3, or 3.	False
7	D is 9, 8, or 7.	True?
8	Exactly two of the digits are 2.	False
9	The clues on days that are factors of 600 are false.	True
10	C is 5.	False
11	E is not 0.	True?
12	Exactly 2 of the digits is/are even.	False
13	E is not 8+1.	True?
14	C is not 3, 9, or 9.	True?
15	C is 4, 4, or 4.	False
16	A is 2 and B is 1.	?
17	A is 2 and B is 2.	?
18	One or more of the digits is 1	True?
19	E is not 5 or 4	True?
20	Exactly 4 of the digits is/are even.	False
21	C is not 2.	True?
22	E is not 7.	True?
23	The clues on day 9 and day 23 are false.	False
24	Strictly less than 10 clues are true.	False

Combining simple clues (assuming the “True?” clues are true, and waiting a bit with clues 8, 12, 16, 17 and 20) gives me C = 7 and D = 9. We also have that E can’t be 2. Only A and B can be 2. Clue 8 tells me, that they can’t both be 2. Therefore clue 17 is false. We can confirm clue 20 is false, as already 2 of the 5 digits are odd. This would mean clue 17 is false, and we arrive at exactly 10 true clues. Time to test that theory.

Day	Clue	True?
1	C is 1, 8, or 0.	False
2	C is 6, 8, or 1.	False
3	A is 5.	False
4	E is 2 or 6.	False
5	D is 3, 7, or 8.	False
6	E is 2, 3, or 3.	False
7	D is 9, 8, or 7.	True
8	Exactly two of the digits are 2.	False
9	The clues on days that are factors of 600 are false.	True
10	C is 5.	False
11	E is not 0.	True
12	Exactly 2 of the digits is/are even.	False
13	E is not 8+1.	True
14	C is not 3, 9, or 9.	True
15	C is 4, 4, or 4.	False
16	A is 2 and B is 1.	True
17	A is 2 and B is 2.	False
18	One or more of the digits is 1	True
19	E is not 5 or 4	True
20	Exactly 4 of the digits is/are even.	False
21	C is not 2.	True
22	E is not 7.	True
23	The clues on day 9 and day 23 are false.	False
24	Strictly less than 10 clues are true.	False

Just using simple clues (not 8 and 12 yet), I get as far as 2179x. This also satisfies clue 8. (I already know E can’t be 2.) The only clue left is clue 12. Combining that with my other knowledge, this means E is 1. So the total solution is ABCDE = 21791. And testing this in the machine confirms it. Merry Christmas!