S1: if it equals S1 + S2 + .... S10, S2 - S10 must collectively equal zero.
S2: # true
S3: true unless # > S1, pretty much guaranteed to be true
S4: # = S4
S5: false if there's a negative number, has to be false
S6: average, but we know that the sum is s1, so we can re-write as s1/6.
S7: true if s4 > s2
S8: S1/S8, which can be rewritten as +/- sqrt(S1)
S9: true if S6 = (s2-s4) - (s8 * s4)
S10: must be a negative integer because when S1 > 36, a negative S8 would decrease more slowly than a positive S6 grew. For numbers < 36, S1 can't be both divisible by 6 and a square root of something.

Working through this we come to the equilibrium situation where:

S1:
S2: 2
S3: true
S4: 2
S5: false
S6:
S7: false
S8:
S9: true
S10:

Otherwise you get into a mess where you have s2 and s4 both equalling 1 which doesn't work, so on so forth. However in order for this situation to exist we need the unlikely constraint of s9 to be true. That means S6 = (S2-S4) - (S8*S4).

Rewriting that,

S1/6 = 0 - ( +/- 2root(S1) ). Because S6 is positive, S8 has to be negative. So, S1/6 = 2sqrt(S1), bring it over, S1/6 - 2sqrt(S1). There's a solution to this equation where S1 = 144. With this, everything falls into place.


S1: 144
S2: 2
S3: true
S4: 2
S5: false
S6: 24
S7: false
S8: -12
S9: true
S10: -16

S1: if it equals S1 + S2 + .... S10, S2 - S10 must collectively equal zero.
S2: # true
S3: true unless # > S1, pretty much guaranteed to be true
S4: # = S4
S5: false if there's a negative number, has to be false
S6: average, but we know that the sum is s1, so we can re-write as s1/6.
S7: true if s4 > s2
S8: S1/S8, which can be rewritten as +/- sqrt(S1)
S9: true if S6 = (s2-s4) - (s8 * s4)
S10: must be a negative integer because when S1 > 36, a negative S8 would decrease more slowly than a positive S6 grew. For numbers < 36, S1 can't be both divisible by 6 and a square root of something.

Working through this we come to the equilibrium situation where:

S1:
S2: 2
S3: true
S4: 2
S5: false
S6:
S7: false
S8:
S9: true
S10:

Otherwise you get into a mess where you have s2 and s4 both equalling 1 which doesn't work, so on so forth. However in order for this situation to exist we need the unlikely constraint of s9 to be true. That means S6 = (S2-S4) - (S8*S4).

Rewriting that,

S1/6 = 0 - ( +/- 2root(S1) ). Because S6 is positive, S8 has to be negative. So, S1/6 = 2sqrt(S1), bring it over, S1/6 - 2sqrt(S1). There's a solution to this equation where S1 = 144. With this, everything falls into place.


S1: 144
S2: 2
S3: true
S4: 2
S5: false
S6: 24
S7: false
S8: -12
S9: true
S10: -16