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One problem - many ways to solve... (Posted on 2017-12-21) Difficulty: 3 of 5
The problem below (Moscow Puzzles #313) can be solved in more than 4 ways,
(each d1 by itself) - using different approaches.

Find the number t and the digit represented by k in:

List your ways of solving it.

No Solution Yet Submitted by Ady TZIDON    
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Solution My ways | Comment 3 of 6 |
1. Trial and error:

Try all 10 possible values of k, either manually with a calculator, with a graphing calculator's table facility, or a spreadsheet, specifying:

a) sqrt(492004 + 100*k)/3 - 230


b) sqrt((492004 + 100*k)/9) - 230

to find the value of t that is integral:

  0                      3.809989331318178                                      
  1                      3.833749108682383                                      
  2                      3.857506472068025                                      
  3                      3.881261422210656                                      
  4                      3.905013959845604                                      
  5                      3.928764085707627                                      
  6                      3.952511800531397                                      
  7                      3.976257105050905                                      
  8                      4                                                      
  9                      4.023740486112047   

so k is 8 and t is 4.

2. More direct:

Take the square root of 492004 and divide by 3. The result is the same 3.809989331318178 seen above; but recognize that the small change that any k would make (the 4th significant figure) is not going to change the result much, so assume the value of t is 4, the next higher digit. Then take [3*(230+4)]^2 verifying that all the given digits match the RHS and k turns out to be 8.

Similarly you could have chosen a starting middle value for k, such as 4 or 5, instead of zero, and likewise found the best digit for t is 4 and proceed from there.

3. Trial and error in the other direction:

Try the LHS with different values for t and see which match the given digits on the RHS. This has the disadvantage of not knowing the approximate size of t--not even that it's necessarily a 1-digit number, so an automated method would be a good choice. But you'd quickly see that method 2 above would be a better choice, as you'd want to know the approximate size of t at least.

  Posted by Charlie on 2017-12-21 11:31:11
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