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Expression Product Exercise (Posted on 2015-09-18) Difficulty: 3 of 5
Prove that when all expessions of the form:
+/- √1 +/- √2 +/- ....... +/- √100 are multiplied together, the result is an integer.
As an example, multiplying all expressions of the form: +/- √1 +/- √2 is equivalent to finding the result of this product:
(√1 +√2)( √1 - √2)( - √1 + √2)( - √1 - √2)

**** Extra Challenge: Solve this puzzle without the aid of a computer program.

No Solution Yet Submitted by K Sengupta    
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even UBASIC can get only to 8 in direct calculation | Comment 4 of 5 |
(In reply to computer exploration -- far from a proof by Charlie)

    5   point 255:dim Term(20)
   10   for N=2 to 10
   20       Prod=1
   25       erase Term()
   30       dim Term(N)
   40       for I=1 to N
   50         Term(I)=sqrt(I)
   60       next
   70       for F=0 to int(2^N-0.5)
   80         Tot=0
   90         Bs="":F1=F
  100         while F1>0
  110           D=F1@2:F1=F1\2
  120           Bs=cutspc(str(D))+Bs
  130         wend
  140         Bs=right("00000000000"+Bs,N)
  150         for I=1 to len(Bs)
  160           if mid(Bs,I,1)="0" then
  170             :Tot=Tot-Term(I)
  180           :else
  190             :Tot=Tot+Term(I)
  200           :endif
  210         next
  220         Prod=Prod*Tot
  230         if N=3 then print using(10,15),Tot
  240       next F
  250       print N,int(Prod+0.5)
  260    next N


 2   1 
 3   64 
 4   4096 
 5   23323703841 
 6   63703464216016403230349121 
 7   316699666163357097153212433469030615484754548657341071360000 
 8   122650145964677485280855470979489605455779538020711656443170899516425755208153578926158095907728782598859083391629280893573529600000000 


Then there's an overflow.

Rounding was used as there were long strings of 9's after the decimal point in each case.

  Posted by Charlie on 2015-09-20 08:23:36
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