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 Thirds and Differences (Posted on 2005-07-21)

Arrange the digits 1-9 so that the first 3 form a number that is 1/3 of the number formed by the last three and the three digits in the middle form a number that is the difference between them.

The numbers are as they appear, not summed.

 No Solution Yet Submitted by Erik O. Rating: 3.0000 (3 votes)

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 My solution | Comment 4 of 9 |

If the 9 digit number is written as ABCDEFGHI then

3ABC = GHI
GHI - ABC = DEF

As GHI is still a 3-digit number then ABC must be less than 333. so A is 1,2 or 3 and G can't be 1 or 2.

If C is 5 then I is 5 but each digit must be different so C & I are not 5.

I then wrote down the numbers 1 to 99 (BC) and then multiplied each number by 3 (HI) and only looking at the last two digits of HI.

I then went through and eliminated the numbers which had duplicates or the number 0 and came up with 34 possibilities:

BC  HI
12  36
16  48
18  54
19  57
21  63
23  69
26  78
27  81
29  87
32  96
38  114
39  117
41  123
43  129
46  138
54  162
58  174
61  183
63  189
64  192
72  216
73  219
76  228
78  234
81  243
82  246
83  249
84  252
86  258
87  261
89  267
91  273
92  276
94  282

I added a third column showing the difference between the two numbers and then went through putting the number 1, 2 & 3 before the two digit number BC and seeing if all three sets of numbers had different digits and I came up with the follwing 4 solutions:

219 438 657
327 654 981
273 546 819
192 384 576

 Posted by Lisa on 2005-07-22 14:09:09

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