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Some digits sum square (Posted on 2011-06-22) Difficulty: 3 of 5
Determine the probability that for a positive integer N drawn at random between 2 and 201 inclusively, the sum of the digits in the base N representation of 2011 (base ten) is a perfect square.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Solution computer solution | Comment 1 of 2

19 out of the 200 numbers in this range satisfy the condition, for a probability of 19/200 = 0.095, or about 1/10.5263157894736842104.


 N
(base)  sod     base-10 representation of the base-N digits
 2       9       1  1  1  1  1  0  1  1  0  1  1
 3       9       2  2  0  2  1  1  1
 8       16      3  7  3  3
 10      4       2  0  1  1
 20      16      5  0  11
 22      16      4  3  9
 26      36      2  25  9
 60      64      33  31
 80      36      25  11
 92      100     21  79
 106     121     18  103
 110     49      18  31
 127     121     15  106
 134     16      15  1
 136     121     14  107
 148     100     13  87
 178     64      11  53
 190     121     10  111
 194     81      10  71
 
stats:
 19      200     19//200         0.095   10.5263157894736842104
 
list
    5   dim Dig(20)
   10     for N=2 to 201
   20       Sod=0:NumDigs=0
   30       Num=2011
   40       while Num>0
   50         D=Num @ N:inc NumDigs:Dig(NumDigs)=D
   60         Sod=Sod+D
   70         Num=Num\N
   80       wend
   90       Sr=int(sqrt(Sod)+0.5)
  100       if Sr*Sr=Sod then inc Psqrs
  101         :print N,Sod,
  102         :for I=NumDigs to 1 step -1
  103           :print Dig(I);
  104         :next:print
  110       inc NCt
  120     next N
  130     print Psqrs,NCt,Psqrs//NCt,Psqrs/NCt,NCt/Psqrs
OK

  Posted by Charlie on 2011-06-22 13:51:00
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