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I've had a couple of goes but end up in contradictions. It is very killer-like! Very enjoyable. Can you post the final solution so I can check I've understood the rules? Perhaps I don't quite understand what "not strictly increasing" means.

Thanks
Ed

Ed

1,2,3,5,8,8,9,13
is an increasing sequence but it is not strictly (or monotonically) increasing.

Clearly this can only apply if the tail is not just in one row/column/nonet.
Solution:
132957468
498623175
576481239
945268713
361794582
287135694
653812947
814579326
729346851

Thanks for the solution. I have all the beginning parts correct which is encouraging. However, I'm a bit confused about how the 2-cell tail at r7c8 fits the definition. Are you sure that tail is correct? If it is I'll have to check the rest of the 2-cell ones again (probably won't be able to get a WT ready until the weekend).

Cheers
Ed

Ed

This is me being tricky: 44 is a non-monotonically increasing sequence summing to 8 - of course it is also non-monotonically decreasing..

Maurice

I just wanted to add some mathematical clarification:

1,2,3,5,8,8,9,13 is a monotonically increasing sequence but it isn't strictly monotonically increasing because of the two 8s.
3,2,1 is a strictly monotonically decreasing sequence.
Therefore 3,3,3 would be a monotonically increasing as well as a monotonically decreasing sequence.

Afmob

Thank you for the clarification, my memory was unreliable here.

For normal usage monotonically appears redundant. However I suppose you could have an increasing sequence where for example "the average of any consecutive ten numbers is greater than the average of any ten consecutive numbers earlier in the sequence" e.g. 10,9,8,7,6,5,4,3,2,1,11,10,9,8,7,6,5,4,3,2,12,11 etc. would be a saw tooth sequence that is increasing in general.

My intention was to keep the possibilities open, hence not strictly.

Cheers

Maurice

Motris 4 WT
long enough so I've not used tiny text
Arrows are listed by the total cell
r1c6=5/6/7/8/9 : r2c7=1/2 r3c8=2/3/4 r4c9 = 2/3/4/5/6
r1c5=2-9 : r2c6=1-8 r3c7=1/2/3/4
r1c4=6/7/8/9 : r1c3=1/2 r1c2=2/3 r2c1=3/4/5/6

r3c1=5-9 : r1c1=1/2/3/4 r2c1 = 4/5/6 (<>3 conflict r1c23)
r2c2=6-9 : r3c1=5/6/7/8 r4c2 = 1/2/3/4

r2c3=689 :r7c5=1=r6c4 r5c3=1/2 r4c4=2 r3c4=3/4

C3: NP r15 12
also a pretty Sue de Coq in R3N3 but it comes out anyway
r4c9<>2-> r1c6<>5

r3c5=3-9 : r3c6=1/2/3/4 r4c6 = 3-9
R3: NQ c4678=1234 -> r1c123r2c1=1234 r2c1=4 123 locked r1
r1c5<>2->r2c6<>1 -> r3c6=1
->2 locked r3c78 -> r2c7=1 -> r3c7 2+ r2c6<>8
5 locked r3c123 Ar3c5: r4c6=5-9
4@r2c1->r1c4<>6

r9c3=7/8/9 : r8c2=1/2 r7c3=3/4 r8c4=3/4/5

This is the conflict move which some might consider a bit too T&E:
Arrow r2c3: if r5c3=2 r4c4=3 r2c3=9 r1c3=1
->r1c1=2/3 -> r3c1=6/7 r4c2=1/3/4 but r2c2<>9 -> r4c2=1
but r2c3=9-> r9c3<>9 -> r8c2=1 conflict C2

r5c3=1 -> r1c3=2 r1c2=3 r1c4=9 r1c1=1 r3c1 =5 r4c2=4 r2c2=9
r4c8=1 r3c9=9 7locked N1 in r3 r3c5=6/8 -> r4c6=5-8
r1c6<>9-> r4c9<>6 r3c8<>4
ir1c5=4-> r2c6r3c7=22-> r3c48=33 r1c5<>4 r2c6<>2
-> r2c5=2 r3c4=4 -> r2c3=8, r3c5=8 -> r4c6=7/8, r1c56<>8

This I consider the good bit:
AR1c5=5/6/7 r2c6=3/5 r3c7=2/3
AR1c6=6/7 r2c7=1 r3c8=2/3 r4c9=3
if r3c8=2 r1c6=6 but r3c7=3 -> r1c5=6
-> r3c78=[23]-> r12c6=[73] -> r4c6=8

r5c2=5/6/8 : r56c1= {23/26} 2 locked C1N4
there is another Sue-de-Coq on C1N7 - not used
8 locked C1r789

r5c5=9 : r56c6={45} r789c6={269}

9 locked N5 r456c5

r3c4=4 -> r8c4<>4 ->r8c2=1 r7c3=3
r9c9=1 r8c3=4 r9c5=4

N7: HP r79c2 ={25} -> Ar5c2=6/8 6 locked N4
C3: HS 6 r3c23=[76]-> NS r6c2=8 r5c2=6 r56c1 ={23}
C2: HS r2c2=6 r1c3=5

r5c8= 2/4/7/8: r6c9=2/4/5/6 r7c8=2/4

r8c6=9 : r9c6=6 r8c7=3
-> r7c6=2 -> r7c8=4 -> r6c9=4 -> r5c8=8
NS from here

I really enjoyed the ideas behind this puzzle. Thanks very much for the work you put into making it. Thanks for posting your WT but I didn't go past the conflict move. I got the wrong impression about this puzzle.

This is what I could find but couldn't get anywhere useful past this. I used some chains including a short conflict chain (step 8,10)

T = tail.
1. T5 at r7c5 = [1+1+1+2+3/1*1*1*2*3/1*1*1*2*4/1+1+2+2+3] = 8/6/9/9
1a. r7c5 = 1, r6c4 = 1, r4c4 = 2
1b. r6c3 = (12), r3c4 = (34)
1c. r2c3 = (689)
1d. r25c3 = [9/1..]

2. T2 at r4c2; [1+8] = [9] blocked by step 1d -> max. r3c1 = 7

3. T3 at r3c1; [3+2+1/3*2*1] blocked by no 1 or 2 left for r1c1 in T2 at r1c1;[5+2+1/6+2+1] blocked by T2 at r1c1 since max. r3c1 = 7 and must also have 1/2 in r1c1
3b T3 at r3c1 = [4+2+1/4*2*1/4+3+1/4+3+2/5+3+1] = 7..9
3c. r2c1 = (45), r1c2 = (23), r1c3 = (12)
3d. r1c4 = (789)

4. naked pair {12} at r15c3: both locked for c3

5. T2 at r1c1. max. r1c1 = 3 since max. r3c1 = 7
5a. min. r3c1 = 5

6. T2 at r4c2. Min. r3c1 = 5 -> max. r4c2 = 4
6a. must be + since r2c2 <> r3c1 -> min r2c2 = 6

7. T3 at r6c2: max. r8c2 = 2 since min. r7c3 & r6c4 = [3+3] = 6
7a. min. r9c3 = 7
7b. max. r7c3 = 4
7c. max. r8c4 = 5

8. no 3 in r1c1 because it forces r14c2 = [21] which is blocked by r8c2. Like this:
8a. 3 in r1c1 -> T2 at r1c1 = [34] = [7] -> T2 at r4c2 = [17] only with r4c2 = 1
8b. 3 in r1c1 -> r1c2 = 2 -> r14c2 = [21] which is blocked.

9. naked pair {12} at r1c13: both locked for r1 & n1
9a. r1c2 = 3
9b. from 3b T3 at r3c1 = [4+3+1/4+3+2/5+3+1] = 8,9 (no 7 in r1c4)

10. r24c2 must be [9/1] -> 9 locked in r2c23 for n1 and r2 and 1 locked in r4c2 + r5c3. Like this.
10a. if 4 in r4c2 -> T2 at r4c2 = [45] = [9] -> r5c3 = 1 (step 1d)
10a. if 1 in r4c2 -> r2c3 = 9 (step 1d)

11. T2 at r3c7, min. r2c6 = 2 since min. r1c5 = 4
11a. r3c6 = 1 (hsingle n2)

12. T2 at r4c6: min. r34c6 = 1*3 -> min. 3c5 = 3
12a. 2 in n2 only in r2: locked for r2

13. T2 at r3c7; max. r3c7 = 4, max. r2c7 = 7

14. T3 at r4c9; min. = 6 -> r1c6 = (6..9)
14a. min. r2c7+r3c8 = 3 -> max. r4c9 = 6
14b.
14c. max. r3c8 + r4c9 = 8 -> max. r2c8 = 4
14d. min. r4c9 = 3
14e. r4c9+r3c8+r2c8 = [3+2+1/3*2*1/3+3+1/3*3*1/4+2+1/4*2*1/4+4+1/ etc
14f. -> r2c7 = 1

15. naked triple {234} in r3c478: all locked for r3 and 2 for n3

16. r2c1 = 4 (hsingle n1)
16a. r3c1 = (56) (T2 at r1c1)
16b. r4c2 + r3c1 = [15] blocked since it r1c1 must be 1 which would leave no 1 for c3
16c. r4c2 + r3c1 = [16]/45] -> r2c2 = (79)

17. 4 in n2 in r1c5 or r3c4 -> no 4 in r3c7 since T2 at r3c7 must be increasing.

18. 5 in n1 only in r3: locked for r3



Cheers
Ed

Ed

I apologise I should have pointed out that I used a conflict move in my solution.

I do not believe/think that you can move forward from your position without it, the unused arrows at this point are just used to finalise the solution.

Maurice