[Guest]

To each positive integer with n² digits, we associate the determinant of a matrix obtained by writing the digits in order across the rows. For example, if n=2, to the integer 5678, we associate determinant of

(56

78) = -2.

Find as a function of n, the sum of all determinants associated with n² digit integers.

Leading digits are assumed to be nonzero, so for n=2, there are 9000 determinants

I found this puzzles at a site so the question may not have been phrased as clearly here. In case you need clarification, pls ask.

[Guest]

please give us another example,

i write a program for n=2

we have 10000 state and just 163 unique number exist!

[Guest]

please give us another example,

i write a program for n=2

we have 10000 state and just 163 unique number exist!

There should be 9000 unique integers for n=2 (4 digit numbers).

Of these there won't be too many distinct determinants associated to integers as you pointed out. because the determinant associated to 5678 would be the same as that for 5768, 8675 and 8765.

Remember, the question is to find the sum of all determinants as a function of n.

[Guest]

we can creat 9999 number for n=4

i write this program with matlab for n=4

sum=0;

for i1=0:9

for i2=0:9

for i3=0:9

for i4=0:9

x=i1*1000+i2*100+i3*10+i4;

if x>999

sum=sum+(det([i1 i2 ;i3 i4]);

end

end

end

end

end

sum

this program generate all number with n=4 (x>999)

calculate sum of determinate of this number , answer is: 20250

i think you question have a bug , please send the source question (you siad : I found this puzzles at a site)

[Guest]

we can creat 9999 number for n=4

i write this program with matlab for n=4

this program generate all number with n=4 (x>999)

calculate sum of determinate of this number , answer is: 20250

i think you question have a bug , please send the source question (you siad : I found this puzzles at a site)

I think the question is ok. There seems to be problem in your program if it gives you 9999 integers with 4 digits!

The problem in your program is that you are taking first digit as zero also.

Besides, the question in the problem is to have the sum of determinants as a function of n. What if n is 5 or 10 or 20 or 50. What is your answer then?

[Guest]

I think the question is ok. There seems to be problem in your program if it gives you 9999 integers with 4 digits!

The problem in your program is that you are taking first digit as zero also.

Besides, the question in the problem is to have the sum of determinants as a function of n. What if n is 5 or 10 or 20 or 50. What is your answer then?

my program is correct with 4 digit , my condition (x>999) reject all number lower than 999 so zero is not a problem

about n=5, 10 ,20,... , i think, first we must solve problem with 4 digit (n=2) then other sample will be solved.

my program said: 20250 and you said 9000!

brother , I don't have any idea !

just see your question again !

i promise ,program is correct , obviously , may be question is wrong.

[Guest]

my program is correct with 4 digit , my condition (x>999) reject all number lower than 999 so zero is not a problem

about n=5, 10 ,20,... , i think, first we must solve problem with 4 digit (n=2) then other sample will be solved.

my program said: 20250 and you said 9000!

brother , I don't have any idea !

just see your question again !

i promise ,program is correct , obviously , may be question is wrong.

I didn't say 9000 was the sum of determinants. The sum of determinants is indeed 20250. 9000 is the number of distinct 4 digit integers.

I didnt use any program, but just some logic and the answer for the sum of determinants as a function of n is 45ⁿx10^(n-1) (which also shows the sum of determinants for n=2 as 20250). May be you should try your program for n=3. You'll find the sum of determinants as 9112500. I never cross-checked the answer as I didnt have it either. May be your program can show whether this answer is correct or not.

[Guest]

I didn't say 9000 was the sum of determinants. The sum of determinants is indeed 20250. 9000 is the number of distinct 4 digit integers.

I didnt use any program, but just some logic and the answer for the sum of determinants as a function of n is 45ⁿx10^(n-1) (which also shows the sum of determinants for n=2 as 20250). May be you should try your program for n=3. You'll find the sum of determinants as 9112500. I never cross-checked the answer as I didnt have it either. May be your program can show whether this answer is correct or not.

now, it's true