perplexus.info :: Just Math : Expression Product Exercise

Expression Product Exercise (Posted on 2015-09-18)

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.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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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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