Three perfect mathematicians with extremely strong memories are taking an exam. The examiner tells each of them a certain piece of information about $x$ and $y$, which two positive integers between and including 2 and 100 one is divisible by the other:
- The first one is told the difference $d=x-y$
- The second one is told the sum $s=x+y$
- The third one the ratio $r=x/y$.
Then the following conversation takes place:
First mathematician: I cannot deduce $x$ and $y$ from their difference $d$.
Second mathematician: I knew that you cannot.
First mathematician: Don't bother to say that, I already knew that you knew that I cannot. But I'm not sure if you don't know that I know that you don't !!!!
Third mathematician: What about me...? I did know the two numbers right away, once they told me the ratio $r$!
Question: What are these mysterious numbers $x$ and $y$?
Notes
- This is the second puzzle that I created on my own, besides the "consecutive-ranks" puzzle.
- I could resume the conversation in just two lines but I preferred to ameliorate it to ease the solution and make it more adapted to logic.
- The solution is unique.
- The uniqueness is twice verified :D
- I will show the solution if this post becomes be sufficiently downvoted or if there are enough unsuccessful attempts
Answer
magic numbers are : (78 , 2 )
First off lets note a_b a couple of two numbers (a*b,b)
Starting from last statement , any set with absent couple such as (2,x) where x > 33 will be excluded
Such couples are:
ratio=35 2-35
ratio=36 2-36
ratio=37 2-37
ratio=38 2-38
ratio=39 2-39
ratio=40 2-40
ratio=41 2-41
ratio=42 2-42
ratio=43 2-43
ratio=44 2-44
ratio=45 2-45
ratio=46 2-46
ratio=47 2-47
ratio=48 2-48
ratio=49 2-49
ratio=50 2-50
So ... standing on that ground , lets begin to work .
this list contains sets of differences that subtractor doesnt know .
set of difference 68 : 2_35 = (70, 35) 4_18 = (72, 18) 17_5 = (85, 5)
set of difference 70 : 2_36 = (72, 36) 5_15 = (75, 15) 7_11 = (77, 11) 10_8 = (80, 8) 14_6 = (84, 6)
set of difference 72 : 2_37 = (74, 37) 3_25 = (75, 25) 4_19 = (76, 19) 6_13 = (78, 13) 8_10 = (80, 10) 9_9 = (81, 9) 12_7 = (84, 7) 18_5 = (90, 5) 24_4 = (96, 4)
set of difference 76 : 2_39 = (78, 39) 4_20 = (80, 20) 19_5 = (95, 5)
set of difference 78 : 2_40 = (80, 40) 3_27 = (81, 27) 6_14 = (84, 14) 13_7 = (91, 7)
set of difference 80 : 2_41 = (82, 41) 4_21 = (84, 21) 5_17 = (85, 17) 8_11 = (88, 11) 10_9 = (90, 9) 16_6 = (96, 6) 20_5 = (100, 5)
set of difference 84 : 2_43 = (86, 43) 3_29 = (87, 29) 4_22 = (88, 22) 6_15 = (90, 15) 7_13 = (91, 13) 12_8 = (96, 8) 14_7 = (98, 7)
set of difference 88 : 2_45 = (90, 45) 4_23 = (92, 23) 8_12 = (96, 12) 11_9 = (99, 9)
set of difference 90 : 2_46 = (92, 46) 3_31 = (93, 31) 5_19 = (95, 19) 6_16 = (96, 16) 9_11 = (99, 11) 10_10 = (100, 10)
set of difference 92 : 2_47 = (94, 47) 4_24 = (96, 24)
set of difference 96 : 2_49 = (98, 49) 3_33 = (99, 33) 4_25 = (100, 25)
this list contains elements grouped by their similar sum , accepted if summer knows for sure that subtractor doesnt know.
set of sum = 66 :
2_34 -temporarily accepted- 5_13 -temporarily accepted- 7_9 -temporarily accepted- 10_6 -temporarily accepted- 14_4 -temporarily accepted-
this list is accepted
set of sum = 68 :
2_35 -temporarily accepted- 3_23 -temporarily accepted- 4_17 -temporarily accepted- 6_11 -temporarily accepted- 8_8 -temporarily accepted- 9_7 -temporarily accepted- 12_5 -temporarily accepted- 18_3 -temporarily accepted- 24_2 -temporarily accepted-
this list is accepted
set of sum = 70 :
2_36 -temporarily accepted-
this list is accepted
set of sum = 72 :
2_37 -temporarily accepted- 4_18 -temporarily accepted- 19_3 -temporarily accepted-
this list is accepted
set of sum = 74 :
2_38 -rejected because 2_38 is unique in the other side of differences-
3_25 6_12 13_5 26_2
this list is rejected
set of sum = 76 :
2_39 -temporarily accepted- 4_19 -temporarily accepted- 5_15 -temporarily accepted- 8_9 -temporarily accepted- 10_7 -temporarily accepted- 16_4 -temporarily accepted- 20_3 -temporarily accepted-
this list is accepted
set of sum = 78 :
2_40 -temporarily accepted-
this list is accepted
set of sum = 80 :
2_41 -temporarily accepted- 3_27 -temporarily accepted- 4_20 -temporarily accepted- 6_13 -temporarily accepted- 7_11 -temporarily accepted- 12_6 -temporarily accepted- 14_5 -temporarily accepted- 21_3 -temporarily accepted- 28_2 -temporarily accepted-
this list is accepted
set of sum = 82 :
2_42 -rejected because 2_42 is unique in the other side of differences-
this list is rejected
set of sum = 84 :
2_43 -temporarily accepted- 4_21 -temporarily accepted- 8_10 -temporarily accepted- 11_7 -temporarily accepted- 22_3 -temporarily accepted-
this list is accepted
set of sum = 86 :
2_44 -rejected because 2_44 is unique in the other side of differences-
3_29 5_17 6_14 9_9 10_8 15_5 18_4 30_2
this list is rejected
set of sum = 88 :
2_45 -temporarily accepted- 4_22 -temporarily accepted- 23_3 -temporarily accepted-
this list is accepted
set of sum = 90 :
2_46 -temporarily accepted-
this list is accepted
set of sum = 92 :
2_47 -temporarily accepted- 3_31 -temporarily accepted- 4_23 -temporarily accepted- 6_15 -temporarily accepted- 8_11 -temporarily accepted- 12_7 -temporarily accepted- 16_5 -temporarily accepted- 24_3 -temporarily accepted- 32_2 -temporarily accepted-
this list is accepted
set of sum = 94 :
2_48 -rejected because 2_48 is unique in the other side of differences-
7_13 14_6
this list is rejected
the previous informations are unnecessary until this point: where subtractor declares his important information:
A list which fits third statement:
set of difference= 68 :
2_35 4_18 17_5 rejected because
it exists a couple (ab,b)=(85,5) in this set where 2_50 is unique in his set of difference where
2_50 has same sum of a_b = 17_5
the summer wudnt say i know that you know that i dont
set of difference= 70 :
2_36 5_15 7_11 10_8 rejected because
it exists a couple (ab,b)=(80,8) in this set where 2_44 is unique in his set of difference where
2_44 has same sum of a_b = 10_8
the summer wudnt say i know that you know that i dont
set of difference= 72 :
2_37 3_25 rejected because
it exists a couple (ab,b)=(75,25) in this set where 2_38 is unique in his set of difference where
2_38 has same sum of a_b = 3_25
the summer wudnt say i know that you know that i dont
set of difference= 76 :
2_39 4_20 19_5
this is the only set accepted because
2_39 : 2_39 has same sum of
5_15 which has same diff of
2_36 which is unique in his set of same sum
4_20 : 4_20 has same sum of
3_27 which has same diff of
2_40 which is unique in his set of same sum
19_5 accepted because there dosnt exist any such unique couple
this set contains the unique solution because there exists two couple 2_39 , 19_5
where the subtractor is not sure whether the summer knows if the subtractor knows that summer dosent know the numbers
set of difference= 78 :
2_40 3_27 6_14 rejected because
it exists a couple (ab,b)=(84,14) in this set where 2_44 is unique in his set of difference where
2_44 has same sum of a_b = 6_14
the summer wudnt say i know that you know that i dont
set of difference= 80 :
2_41 4_21 5_17 rejected because
it exists a couple (ab,b)=(85,17) in this set where 2_44 is unique in his set of difference where
2_44 has same sum of a_b = 5_17
the summer wudnt say i know that you know that i dont
set of difference= 84 :
2_43 3_29 rejected because
it exists a couple (ab,b)=(87,29) in this set where 2_44 is unique in his set of difference where
2_44 has same sum of a_b = 3_29
the summer wudnt say i know that you know that i dont
set of difference= 88 :
2_45 : 2_45 has same sum of
23_3 which has same diff of
46_2 which is unique in his set of same sum
4_23 : 4_23 has same sum of
3_31 which has same diff of
2_46 which is unique in his set of same sum
8_12 : 8_12 has same sum of
13_7 which has same diff of
2_40 which is unique in his set of same sum
11_9 : 11_9 has same sum of
10_10 which has same diff of
2_46 which is unique in his set of same sum
rejected because all the couples in this set leads to atleast one unique couple in the list of addition
which means subtractor knows cetainly that summer doenst know if subtractor knows wheather summer doesnt know numbers
set of difference= 90 :
2_46 3_31 5_19 6_16 rejected because
it exists a couple (ab,b)=(96,16) in this set where 2_50 is unique in his set of difference where
2_50 has same sum of a_b = 6_16
the summer wudnt say i know that you know that i dont
set of difference= 92 :
2_47 : 2_47 has same sum of
3_31 which has same diff of
2_46 which is unique in his set of same sum
4_24 : 4_24 has same sum of
5_19 which has same diff of
2_46 which is unique in his set of same sum
rejected because all the couples in this set leads to atleast one unique couple in the list of addition
which means subtractor knows cetainly that summer doenst know if subtractor knows wheather summer doesnt know numbers
set of difference= 96 :
2_49 3_33 rejected because
it exists a couple (ab,b)=(99,33) in this set where 2_50 is unique in his set of difference where
2_50 has same sum of a_b = 3_33
the summer wudnt say i know that you know that i dont
from the last list there s just one accepted set , where there s just one accepted couple specified from last divisor's declaration.
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