The Missing Number(s)

Using the numbers 1-9 at most one time ( i.e. there is 1 number 1, 1 number 2, 1 number 3 ... to choose from )

One number isn't in anyone's box, which numbers(s) could that be?

Statements:

Farrah: "We all have two numbers each, and one of us has the number 1"

Kelly: "One of my numbers is the square of Farrah's smallest number"

Letoya: "My numbers are the digits of Kelly's product"

Michelle: "My numbers are the digits of Letoya's product"

@joe-shmo said
Using the numbers 1-9 at most one time ( i.e. there is 1 number 1, 1 number 2, 1 number 3 ... to choose from )

One number isn't in anyone's box, which numbers(s) could that be?

Statements:

Farrah: "We all have two numbers each, and one of us has the number 1"

Kelly: "One of my numbers is the square of Farrah's smallest number"

Letoya: "My numbers are the digits of Kelly's product"

Michelle: "My numbers are the digits of Letoya's product"

Is it possible that the System is underdefined?

Let the tiles be F1,F2,K1,K2,L1,L2,M1,M2

F's Statement: one Person has the 1
K's Statement K1=F1^2, thus K1=4 or 9 and F1 =2 or 3 F2>F1, thus F1<>1
L's Statement: K1*K2=L1L2
If K1=4 then K2=9 (since then F1=2 and 12,20,24,28 and 32 are impossible for K1*K2 since all tiles are given only once)
M's Statement L1*L2=M1M2 thus M1=1 M2= 8
Thus F2= 5 or 7 both of which hold for the original condition.

If K1=9 (F1=3)
K2=2 => L1L2=18 then M1M2=08 (= not included)
K2=4 => L1L2=36 but F1=3
K2=5 => L1L2=45 then M1M2=20 (0 not included)
K2=6 => L1L2=54 see above
K2=7 => L1L2=63 F1=3
K2=8 => L1L2= 72 => M1M2=14 then F2=5 or F2=6 in accordance with original condition

@ponderable said
Is it possible that the System is underdefined?

Let the tiles be F1,F2,K1,K2,L1,L2,M1,M2

F's Statement: one Person has the 1
K's Statement K1=F1^2, thus K1=4 or 9 and F1 =2 or 3 F2>F1, thus F1<>1
L's Statement: K1*K2=L1L2
If K1=4 then K2=9 (since then F1=2 and 12,20,24,28 and 32 are impossible for K1*K2 since all tiles are given only once)
M's Statement L1*L2=M1M ...[text shortened]... => L1L2=63 F1=3
K2=8 => L1L2= 72 => M1M2=14 then F2=5 or F2=6 in accordance with original condition

Underdefined as in multiple solutions? Yes. There a 4 solutions, and I believe you have them, short of making the final statement. The only numbers that could be missing among all the solutions are 5,6,and 7.

@joe-shmo said
Underdefined as in multiple solutions? Yes. There a 4 solutions, and I believe you have them, short of making the final statement. The only numbers that could be missing among all the solutions are 5,6,and 7.

Yes that was what I meant (no singular solution).

So the missing number can be 5 or 7 if F1=2 and 5 or 6 if F1=3.

That F1 is the smaller number is actually not narrowing down things.

I started by nailing down who had the "1".

1) Assume Farrah has the "1", which is guaranteed to be her smallest number.
Thus, Kelly would have to have the square of it which is also "1", which they both can't have. Thus Farrah does not have the "1".

2) Kelly cant have the "1" because Latoya must have two digits from Farrah's statement. The largest product that could be formed is a single digit (1*9 = 9 ).

Letoya cant have the "1" by the same reasoning, but with Michelle. Thus, Michelle has the "1"

From here I just did case work checking that her numbers are a product that has distinct factors < 10. Michelle could have:

1,1 - (11 Prime factor larger than 10 & also would imply two "1's" )
1,2
1,3 - (13 Prime factor larger than 10)
1,4
1,5
1,6
1,7 (17- Prime factor larger than 10)
1,8
1,9 (19 - Prime factor larger than 10)
2,1
3,1 (31 - Prime factor larger than 10)
4,1 (41 - Prime factor larger than 10)
5,1 (51 - Prime factor larger than 10)
6,1 (61 - Prime factor larger than 10)
7,1 (71 - Prime factor larger than 10)
8,1 (81 = 9*9 = non distinct factors)
9,1 (91 - Prime factor larger than 10)

Then just check each case:

Michelle 1,2

Letoya 2,6 - Not allowed- multiple 2's
..................3,4 - Products 34 = 17*2 or 43 = 43*1

This line is not possible, Michelle cannot have 1,2.

Then just repeat this process for the remaining possibilities of Michelle and find the lines that have possible solutions are Michelle has 1,4 or 1,8.

If anyone wishes to see the specific contradictions that arise I'll type them up. Otherwise, good solve Ponderable!

@joe-shmo said
Using the numbers 1-9 at most one time ( i.e. there is 1 number 1, 1 number 2, 1 number 3 ... to choose from )

One number isn't in anyone's box, which numbers(s) could that be?

Statements:

Farrah: "We all have two numbers each, and one of us has the number 1"

Kelly: "One of my numbers is the square of Farrah's smallest number"

Letoya: "My numbers are the digits of Kelly's product"

Michelle: "My numbers are the digits of Letoya's product"

I haven't looked at Pondies answer below but I think the number must be either 5 or 6
Farah -3 and 5 or6
Kelly -8 and 9(3 squared =9)
Liota 2 and 7(8 *9 =72)
Michelle 1 and 4(2*7 =14)

@venda said
I haven't looked at Pondies answer below but I think the number must be either 5 or 6
Farah -3 and 5 or6
Kelly -8 and 9(3 squared =9)
Liota 2 and 7(8 *9 =72)
Michelle 1 and 4(2*7 =14)

5 or 6 are correct, but your missing 1 more number.

@joe-shmo said
5 or 6 are correct, but your missing 1 more number.

I see now from Pondies posts that there is more than one solution.
I didn't realise there was more than one answer
I did apply some logic when coming up with my one solution but in nowhere as much detail.
Thanks for the puzzle anyway.
Anything to get the brain cells working these days is welcome.