Number 100

[mfgoode]

Excellent work Dario and your factorisation applet is superb. I must put good use to it
You are well worth your salt.
Keep up the good work and all the best.
Mally

[mfgoode]

Originally Posted by fetofs

Mally is talking about mixed numbers, of the form ; I don't see why we should allow parentheses here....

On a side note, I bet Lucas didn't have a computer!


94+1578/263 = 100
91+7524/836 = 100
3+69258/714 = 100
91+5823/647 = 100
96+1428/357 = 100
96+2148/537 = 100
82+3546/197 = 100
81+5643/297 = 100
81+7524/396 = 100
96+1752/438 = 100
91+5742/638 = 100

It is pretty obvious that not only number 100 but also all numbers up to 100 excepting 1,2,3,4, can be written by mixed fractions with the same rules
Eg : 9 + 5472/1368 = 13.
The tough nuts to crack are 15 and 18 though these may be solved as a simple fraction.
To stick to mixed fractions we must assume a whole number +..x/y using all the 9 digits. There is a way to adhere to this rule.
Can you give it? I mean the mixed fractions for 15 and 18?
Mally

[R. Gerbicz]

Originally Posted by mfgoode

It is pretty obvious that not only number 100 but also all numbers up to 100 excepting 1,2,3,4, can be written by mixed fractions with the same rules

I've computed all solutions up to 100 using the following (PARI) program:

sol=vector(100,i,0);A=matrix(100,300);b=c=vector(9);for(n=1,9!,u=numtoperm(9,n);a=0;\
x=0;for(i=1,9,x=10*x+u[10-i];c[10-i]=x);for(i=0,2,x=0;for(j=i+1,9,x=10*x+u[j];b[j]=x);\
for(j=i+1,8,x=a+b[j]/c[j+1];if((x<=100)&&(type(x)=="t_INT"),sol[x]++;y=sol[x];\
A[x,3*y-2]=a;A[x,3*y-1]=b[j];A[x,3*y]=c[j+1]));a=10*a+u[i+1]));\
for(i=1,100,for(j=1,sol[i],write("mixed.txt",i,"=",A[i,3*j-2],"+",A[i,3*j-1],"/",A[i,3*j])))

See mixed.txt for the solutions.

[fetofs]

Originally Posted by R. Gerbicz

I've computed all solutions up to 100 using the following (PARI) program:

sol=vector(100,i,0);A=matrix(100,300);b=c=vector(9);for(n=1,9!,u=numtoperm(9,n);a=0;\
x=0;for(i=1,9,x=10*x+u[10-i];c[10-i]=x);for(i=0,2,x=0;for(j=i+1,9,x=10*x+u[j];b[j]=x);\
for(j=i+1,8,x=a+b[j]/c[j+1];if((x<=100)&&(type(x)=="t_INT"),sol[x]++;y=sol[x];\
A[x,3*y-2]=a;A[x,3*y-1]=b[j];A[x,3*y]=c[j+1]));a=10*a+u[i+1]));\
for(i=1,100,for(j=1,sol[i],write("mixed.txt",i,"=",A[i,3*j-2],"+",A[i,3*j-1],"/",A[i,3*j])))

I hereby officially present R. Gerbicz with the award of making the most extremely obscure programs!

[mfgoode]

Originally Posted by R. Gerbicz

I've computed all solutions up to 100 using the following (PARI) program:

sol=vector(100,i,0);A=matrix(100,300);b=c=vector(9);for(n=1,9!,u=numtoperm(9,n);a=0;\
x=0;for(i=1,9,x=10*x+u[10-i];c[10-i]=x);for(i=0,2,x=0;for(j=i+1,9,x=10*x+u[j];b[j]=x);\
for(j=i+1,8,x=a+b[j]/c[j+1];if((x<=100)&&(type(x)=="t_INT"),sol[x]++;y=sol[x];\
A[x,3*y-2]=a;A[x,3*y-1]=b[j];A[x,3*y]=c[j+1]));a=10*a+u[i+1]));\
for(i=1,100,for(j=1,sol[i],write("mixed.txt",i,"=",A[i,3*j-2],"+",A[i,3*j-1],"/",A[i,3*j])))

See mixed.txt for the solutions.

Excelent R. Gerbicz
However the number 15 you have given as 15=0+27945/1863 but this is not a mixed fraction which is of the form a + b/c so 0 is not permissible as it does not qualify as a whole number.

The other is 18 in the same form but you have omitted it altogether!

Hint: Your computer and PARI program is right no doubt and I am merely pointing for a gimmicky way of writing the mixed fraction which IS a complex fraction but it qualifies as a mixed fraction.
Please try it out and dont swear at me as it is so very simple!
Mally

[retina]

Originally Posted by mfgoode

0 is not permissible as it does not qualify as a whole number.

Strange, I was taught that whole numbers started at 0 and natural numers started at 1.

[alpertron]

retina is right. Please see this Wikipedia article about whole number.

[mfgoode]

Originally Posted by retina

Strange, I was taught that whole numbers started at 0 and natural numers started at 1.

Good point Retina!

However I made it very clear in my previous post on distinguishing between a simple fraction b/c and a mixed fraction a +b/c. If you make a=0 then there will be no difference. And here I mean that 'a' has a value and NOT zero.

To avoid ambiguity I used the old term which was in use before the idea of sets and the empty set was conceived when the zero came into prominence.

So lets not quibble over definitions, neither should one be dogmatic, Alpertron, when the solution of the problem is what is required.

And in this regard my hint is that the solution, by the way, is a complex fraction such as a+ (b)/(c/d). If I do go further I will give the game away but I will, if R.Gerbicz does not arrive at the solution. He deserves it after the pains he has taken over it.

Well lets have the final word from Wikipedia which we all seem to follow these days. No wonder Wiki has an edit section!

"Peano axioms. There are many systems that satisfy these axioms, including the natural numbers (either starting from zero or one)".[extract from Wiki]

I hope that this is clear but I will welcome any negative suggestions on the non negative integers where zero is included to distinguish it from the Natural or whole numbers now known as the positive integers.

I thank you and Alpertron to bring this delicate point up and in this case it is worth splitting hairs.

Mally

[alpertron]

This time it appears that Mally is right. The original problem in post #1 was:

Originally Posted by mfgoode

The Problem: To write 100 using all 9 digits (1 -- 9) in the form of a mixed number; Eg: 100 = 91 + 5742/638 or 91(5742/638) a mixed number and no repitiion of digits but just the nine.

So in this extension (replacing the value 100 by other numbers) no zeros are allowed, only the digits 1 to 9 with no repetition. This invalidates all solutions of the form 0 + abcde/fghi .

[R. Gerbicz]

Originally Posted by mfgoode

And in this regard my hint is that the solution, by the way, is a complex fraction such as a+ (b)/(c/d).

OK, I've computed all "complex" solutions up to 100 by the following (PARI) program

sol=vector(100,i,0);A=matrix(100,1000);b=matrix(9,9);for(n=1,9!,u=numtoperm(9,n);\
for(i=1,9,x=0;for(j=i,9,x=10*x+u[j];b[i,j]=x));\
for(i=1,2,for(j=i+1,7,for(k=j+1,8,x=b[1,i]+b[i+1,j]/(b[j+1,k]/b[k+1,9]);\
if((x<=100)&&(type(x)=="t_INT"),sol[x]++;y=4*sol[x];\
A[x,y-3]=b[1,i];A[x,y-2]=b[i+1,j];A[x,y-1]=b[j+1,k];A[x,y]=b[k+1,9])))));\
for(i=1,100,for(j=1,sol[i],y=4*j;\
write("complex.txt",i,"=",A[i,y-3],"+",A[i,y-2],"/(",A[i,y-1],"/",A[i,y],")")))

You can see the solutions in complex.txt file.

Originally Posted by alpertron

This invalidates all solutions of the form 0 + abcde/fghi .

Yes, you're right.

[mfgoode]

Well R.Gerbicz you have it got it right this time. Congratulations!
Thanks Alpertron for elucidating my point on the mixed fractions.
The specific ones I had for 15 and 18 are
15 = 3 + 8952/746/1 and yours is even better 3 + 1/746/8952.
The other was
18 = 9 + 5742/638/1 which you have given as 9 + 1/638/5742
Your attachment giving all the values is amazing and gives me the incentive that I could put my computer to better use.
I hope to get another p.c. which I may devote to joining in primes search and the various kinds you people post about.
Thank you, and I think we could put a lid on this thread.
Mally