X=14 for"聪明人都来做做：黑板上的数字X是什么？"

salmonfish

www.ddhw.com

For all possible values of ABC and A+B+C:
X ≠ the following unique values,. Otherwise, A, B, C should have known the answer even before any trying:
720=8x9x10; 630=7x9x10; 560=7x8x10; 504=7x8x9; 540=6x9x10; 480=6x8x10; 432=6x8x9; 420=6x7x10; 378=6x7x9; 336=6x7x8; 450=5x9x10; 400=5x8x10; 350=5x7x10; 315=3x7x9; 300=5x6x10; 320=4x8x10; 288=4x8x9; 252=4x7x9; 224=4x7x8; 192=4x6x8; 200=4x5x10; 270=3x9x10; 189=3x7x9; 62=3x6x9; 150=3x5x10; 135=3x5x9; 105=3x5x7; 112=2x7x8; 100=2x5x10; 64=2x4x8; 63=1x7x9:
50=1x5x10; 45=1x5x9; 35=1x5x7; 32=1x4x8; 28=1x4x7; 27=1x3x9; 6=1x2x3=1+2+3; 7=1+2+4; 26=7+9+10www.ddhw.com

First round:
A: I don’t know.
Indicates:
• A ≠ 8. Otherwise, must B=9, C=10, and X= 720=8x9x10 or X= 27=8+9+10
• A ≠ 7. otherwise, when
(1) B=8, C=9, =>X= ABC=504 (unique), or X= A+B+C=24
(2) B=8, C=10, =>X=ABC=560 (unique), or X= A+B+C=25
(3) B=9,C=10, => X=ABC=630 (unique), or X= A+B+C=26 (unique)
So A=1, or 2, or 3, or 4, or 5, or 6 with conditions:
• X≠ 36. Otherwise, for 36=ABC=2x3x6=1x4x9, A should know
• X≠ 42. Otherwise, for 42=ABC=2x3x7=1x6x7, A should know
• X≠ 54. Otherwise, for 54=ABC=2x3x9=1x6x9, A should know
• X≠ 56. Otherwise, for 56=ABC=2x4x7=1x7x8, A should know
• X≠ 70. Otherwise, for 70=ABC=2x5x7=1x7x10, A should know
• X≠ 72. Otherwise, for 72=ABC=3x4x6=1x8x9, A should know
• X≠ 84. Otherwise, for 84=ABC=3x4x7=2x6x7, A should know
• X≠ 90. Otherwise, for 90=ABC=3x5x6=2x5x9=1x9x10, A should know
• X≠ 96. Otherwise, for 96=ABC=3x4x8=2x6x8, A should know
• X≠ 108. Otherwise, for 108=ABC=3x4x9=2x6x9, A should know
• X≠ 120. Otherwise, for 120=ABC=4x5x6=3x4x10=2x6x10, A should know
• X≠ 126. Otherwise, for 126=ABC=3x6x7=2x7x9, A should know
• X≠ 140. Otherwise, for 140=ABC=4x5x7=2x7x10, A should know
• X≠ 144. Otherwise, for 144=ABC=3x6x8=2x8x9, A should know
• X≠ 160. Otherwise, for 160=ABC=4x5x8=2x8x10, A should know
• X≠ 168. Otherwise, for 168=ABC=4x6x7=3x7x8, A should know
• X≠ 180. Otherwise, for 180=ABC=4x5x9=3x6x10=2x9x10, A should know
• X≠ 210. Otherwise, for 210=ABC=5x6x7=3x7x10, A should know.
• X≠ 216. Otherwise, for 216=ABC=4x6x8=3x8x9, A should know
• X≠ 240. Otherwise, for 200=ABC=5x6x8=4x6x10 =3x8x10, A should know
• X≠ 270. Otherwise, for 270=ABC=5x6x9=3x9x10, A should know
• X≠ 280. Otherwise, for 280=ABC=5x7x8=4x7x10, A should know
• X≠ 360. Otherwise, for 360=ABC=5x8x9=4x9x10, A should know
• X≠ 25. Otherwise, for 25=A+B+C=6+9+10, A should know
•
B: I don’t know. (based on A’s answer, and knowledge of the unique numbers)
Indicates:
• X≠ 80, for 80=ABC= 2x4x10=2x5x8=1x8x10, If A held 1, A should have known. If A held 2, A should have not known, then B should have known.
• X≠ 60, for 60=ABC=3x4x5=1x6x10=2x3x10=2x5x6, If A held 3 or 1, A should have known. If A held 2, A should have not known, then B should have known.
• X≠ 48, for 48=ABC= 2x4x6=2x3x8=1x6x8, If A held 1, A should have known. If A held 2, A should have not known, then B should have known.
• X≠ 40, for 40=ABC= 1x5x8=1x4x10=2x4x5, If A held 2, A should have known. If A held 1, A should have not known, then B should have known.
• X≠ 30, for 30=ABC= 2x3x5=1x5x6=1x3x10, If A held 2, A should have known. If A held 1, A should have not known, then B should have known.
• X≠ 24, for 24=ABC= 2x3x4=1x3x8=1x4x6, and 24=A+B+C= 6+8+10=5+9+10, If A held 2, 5, or 6, A should have known. If A held 1, A should not have known, then B should have known.
• X≠ 23, for 23=A+B+C= 4+9+10=5+8+10=6+7+10=6+8+9, If A held 4, or 5, A should have known. If A held 6, A should not have known, then B should have known.
• X≠ 22, for 22=A+B+C=3+9+10= 4+8+10=6+7+9=5+7+10=5+8+9, If A held 3, 4, or 6, A should have known. If A held 5, A should have not known, then B should have known.
• X≠ 9, for 9=A+B+C=2+3+4= 1+2+6=1+3+5, If A held 2, A should have known. If A held 1, A should have not known, then B should have known.www.ddhw.com

C: I don’t know. (based on A and B's answers, and knowledge of the unique numbers)
Indicates:
• X≠ 21, for 21=1x3x7=2+8+9= 6+7+8=4+7+10=4+8+9=5+7+9=5+6+10, If A held 1, 2 or 6, A should have known. If A held 4 or 5, B held 6 or 8, A should have not known, B should have known If A held 4 or 5, B held 7, A and B should have not known, C should have known.
• X≠ 10, for 10=1x2x5=1+2+7= 1+3+6=1+4+5=2+3+5, If A held 2, A should have known. If A held 1, B held 3 or 4, A should have not known, B should have known, If A held 1, B held 2, A and B should have not known, C should have known
• X≠ 8, for 8=1x2x4=1+2+5= 1+3+4, If A held 1, B held 3 , A should not know, B should know, If A held 1, B held 2, A and B should have not known, C should have known.www.ddhw.com

(After 1st round, among all the possible values of ABC or A+B+C, only 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 left.) Following analysis combines 1st round and 2nd round answers:
• X≠ 20.
For 20=
2+8+10=3+8+9=3+7+10=4+7+9=4+6+10=5+6+9=5+7+8=1+9+10=1x4x5=1x2x10
1st round:
If A held 2, A should have knownthe answer. So A should hold 1, 3, 4 or 5.
For A held 1, 3, 4 or 5, if B held 2, 4, 8, or 9, B should have known the answer. So B should hold either 6 or 7.
For B held 6 or 7, if C held 8, C should have known the answer. So C held 9 or 10.
2nd round:
For C held 9 or 10, A should know the answer.www.ddhw.com

• X≠ 19.
For 19=1+8+10=2+7+10= 3+6+10=3+7+9=4+5+10=4+6+9=4+7+8=5+6+8
1st round:
If A held 1, 2 or 5, A should have known the answer. So A should hold 3 or 4.
For A held 3 or 4, if B held 5, B should have known the answer. So B should hold either 6 or 7.
For B held 6 or 7, if C held 8 or 10, C should have known the answer, So C should hold 9.
2nd round:
For C held 9, A should have known the answer.www.ddhw.com

• X≠ 18,
For 18=5+6+7=4+5+9= 3+5+10=3+6+9=3+7+8=2+6+10=2+7+9=1+7+10=1+8+9=
=1x2x9=1x3x6,
1st round:
If A held 5, A should have known the answer. So A held 1, 2, 3 or 4.
For A held 1, 2, 3 or 4, if B held 2, 3 or 8, B should have known the answer. So B should hold 5, 6 or 7.
For B held 5, 6 or 7, if C held 8, 9 or 10, C should have no clue.
2nd round and 3rd round:
A, B, C all had no further clue,www.ddhw.com

• X≠ 17,
For 17=4+5+8=4+6+7= 3+4+10=3+5+9=3+6+8=2+5+10=2+6+9=2+7+8=1+6+10=
=1+7+9
1st round:
A had no clue
If B held 4, B should have known the answer. So B should hold 5, 6 or 7.
For B held 5, 6 or 7, if C held 7, C should have known the answer. So C held 8, 9,10.
2nd round and 3rd round:
For C held 8, 9, 10, A, B, C all had no further clue,

• X≠ 16,
For 16=4+5+7=3+4+9= 3+5+8=3+6+7=2+4+10=2+5+9=2+6+8=1+5+10=
=1+6+9 =1+7+8
1st round:
If A held 4, A should have known the answer. So A held 1, 2, or 3.
For A held 1, 2, or 3, if B held 7, B should have known the answer. So B should hold 4, 5 or 6.
For B held 4, 5 or 6, if C held 7, C should have known the answer. So C held 8, 9, 10.
2nd round and 3rd round:
For C held 8, 9, 10, A, B, C all had no further clue,www.ddhw.com

• X≠ 15,
For 15=4+5+6=3+4+8= 3+5+7=2+3+10=2+4+9=2+5+8=2+6+7=1+4+10=
=1+5+9 =1+6+8=1x3x5
1st round:
If A held 4, A should have known the answer. So A held 1, 2, or 3.
For A held 1, 2, or 3, B had no clue, and if C held 5, C should have known the answer. So C should hold 7, 8, 9 or 10.
2nd round
For C held 7, 8, 9 or 10, A had no clue, if B held 3, B should have known the answer. So B held 4, 5, 6.
For B held 4, 5, 6, if C held 10, C should have known the answer. So C held 7, 8, or 9.
3rd round:
For C held 7, 8, or 9, A, B, C all had no further clue.www.ddhw.com

• X≠ 13,
For 13=3+4+6=2+3+8= 2+4+7=2+5+6=1+2+10=1+3+9=1+4+8=1+5+7,
1st round:
If A held 3, A should have known the answer. So A held 1 or 2.
For A held 1 or 2, if B held 2, B should have known the answer. So B should hold 3, 4 or 5.
For B held 3, 4 or 5, if C held 6 or 9, C should have known the answer. So C should hold 7 or 8.
2nd round:
For C held 7 or 8, if B held 3 or 5, B should have known the answer. So B should held 4
For B held 4, C should have known the answer.www.ddhw.com

• X≠ 12,
For 12=3+4+5=2+3+7= 2+4+6=1+2+9=1+3+8=1+4+7=1+5+6=1x2x6=1x3x4,
1st round:
If A held 3, A should have known the answer. So A held 1 or 2.
For A held 1 or 2, if B held 5, B should have known the answer. So B should hold 3, 3 or 4.
For B held 2, 3 or 4, if C held 4, 8 or 9, C should have known the answer. So C should hold 6 or 7.
2nd round:
For C held 6 or 7, if B held 2 or 3, B should have known the answer. So B should held 4
For B held 4, C should have known the answer.www.ddhw.com

• X≠ 11,
For 11=1+2+8=1+3+7= 1+4+6=2+3+6=2+4+5,
1st round:
For A held 1 or 2, if B held 2, B should have known the answer. So B should hold either 3 or 4.
For B held 3 or 4, if C held 5 or 7, C should have known the answer.
So C should hold 6,
2nd round:
For C held 6, A should have known the answer.

Among all possible values of ABC and A+B+C, only X=14 left, and
X=14. For 14=3+4+7=3+5+6=2+3+9=2+4+8=2+5+7=1+3+10=1+4+9=1+5+8=1+6+7=1x2x7
1st round:
A had no clue.
If B held 2 or 6, B should have known the answer. So B held 3, 4, or 5.
For B held 3, 4, or 5, If C held 6 or 10, C should have known the answer. So C held 7, 8 or 9.
2nd round:
For C held 7, 8, or 9, if A held 3, A should have known the answer. So A held 1, or 2.
For A held 1 or 2, if B held 3, or 6, B should have known the answer. So B held 4 or 5.
For B held 4 or 5, if C held 7 or 9, C should have known the answer. So C held 8.
3rd round:
For C held 8, A, holding 1 or 2, should have known the answer.

 

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  本贴由[salmonfish]最后编辑于：2009-1-19 20:55:32  

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