A new December and a new bunch of puzzles from mscroggs.co.uk.

All the hints:

Day	Clue
1	144 is an elf’s three-digit number.
2	The first elf’s one-digit number is not a factor of 202.
3	The first elf’s one-digit number is not 7.
4	The third elf’s one-digit number is not 9.
5	The second elf’s one-digit number is not 1, 7, or 9.
6	990 is a multiple of an elf’s three-digit number.
7	The third elf’s one-digit number is not 1.
8	The third elf’s one-digit number is not a factor of 432.
9	The first elf’s one-digit number is not 5.
10	The second elf’s one-digit number is not 4, 9, or 5.
11	The third elf’s one-digit number is not 2.
12	The second elf’s one-digit number is not 2, 8, or 1.
13	One of the digits of the second elf’s three-digit number is 9.
14	The second elf’s one-digit number is not 6, 2, or 5.
15	The third elf’s one-digit number is not 7.
16	The first elf’s one-digit number is not 3.
17	The first elf’s one-digit number is not 9.
18	The first elf’s one-digit number is not 6.
19	The highest common factor of 256 and the second elf’s three-digit number is 1.
20	The third elf’s one-digit number is not 4.
21	138 is an elf’s three-digit number.
22	The highest common factor of 851 and the third elf’s three-digit number is 1.
23	The highest common factor of the first and third elves’ one-digit numbers is not 2, 4, or 1.
24	Santa’s number is 444.

And I know no. 8 and 23 are wrong.

My first look at this problem:

• (aaa – bbb)*c = x
• (x – ddd)*e = y
• (y – fff)*g = hhhhh

Let’s sort this a little bit.

Day	Clue
24	Santa’s number is 444.

Very clear, and as I suspected, very late in the game. aaa = 444.

Day	Clue
1	144 is an elf’s three-digit number.
6	990 is a multiple of an elf’s three-digit number.
13	One of the digits of the second elf’s three-digit number is 9.
19	The highest common factor of 256 and the second elf’s three-digit number is 1.
21	138 is an elf’s three-digit number.
22	The highest common factor of 851 and the third elf’s three-digit number is 1.

Or to put it another way:

• We’re looking for bbb, ddd and fff.
• One of these is 144.
• One of these is 138. (6 * 23)
• One of these is a multiple of 990, therefore one of these: 110, 165, 198, 330, 495, 990. There’s no overlap with the 2 we already have, so this must be the third one.
• One of the digits of ddd is 9. This must be the “multiple of 990” one.
• As 256 is 2⁸, ddd must be odd.
• Combining these 3 facts, we get ddd = 495.
• As 851 = 23 * 37, fff can’t be 138. fff = 144.
• bbb = 138.

Day	Clue
2	The first elf’s one-digit number is not a factor of 202.
3	The first elf’s one-digit number is not 7.
9	The first elf’s one-digit number is not 5.
16	The first elf’s one-digit number is not 3.
17	The first elf’s one-digit number is not 9.
18	The first elf’s one-digit number is not 6.

c can’t be 2, 7, 5, 3, 9 or 6. This leaves 1, 4 and 8. (I assume for now, that none of the 1 digit numbers are 0.)

Day	Clue
5	The second elf’s one-digit number is not 1, 7, or 9.
10	The second elf’s one-digit number is not 4, 9, or 5.
12	The second elf’s one-digit number is not 2, 8, or 1.
14	The second elf’s one-digit number is not 6, 2, or 5.

e can’t be 1, 7, 9, 4, 5, 2, 8, 6. This leaves 3.

Day	Clue
4	The third elf’s one-digit number is not 9.
7	The third elf’s one-digit number is not 1.
11	The third elf’s one-digit number is not 2.
15	The third elf’s one-digit number is not 7.
20	The third elf’s one-digit number is not 4.

g can’t be 9, 1, 2, 7 or 4. This leaves 3, 5, 6 and 8.

Let’s repeat that:

• (aaa – bbb)*c = x
• (x – ddd)*e = y
• (y – fff)*g = hhhhh
• aaa = 444
• bbb = 138
• c is 1, 4 or 8
• ddd = 495
• e = 3
• fff = 144
• g is 3, 5, 6 or 8

Now it’s just a question of going through all the options. Discarding the negative options for hhhhh first, and testing the true 5 digit options next.

444	138	1	495	3	144	3	-2133
444	138	4	495	3	144	3	6129
444	138	8	495	3	144	3	17145	no
444	138	1	495	3	144	5	-3555
444	138	4	495	3	144	5	10215	no
444	138	8	495	3	144	5	28575	no
444	138	1	495	3	144	6	-4266
444	138	4	495	3	144	6	12258
444	138	8	495	3	144	6	34290	no
444	138	1	495	3	144	8	-5688
444	138	4	495	3	144	8	16344	ja
444	138	8	495	3	144	8	45720	no

Hey! Turns out, (((444-138)*4-495)*3-144)*8 = 16344.