Online Course Discussion Forum

MCIII Number Theory 2.30 and 2.32

 
 
JinTina的头像
MCIII Number Theory 2.30 and 2.32
JinTina - 2023年12月27日 Wednesday 18:53
 

Hello,


I don't know how to solve 2.30. I tried giving the 9 numbers in the square a variable and summed the equations but I don't know how to solve it.

For 2.32 I just tried to pair consecutive numbers and subtract them, after one round I get: 1, 1, 1, 1, 1,... 1, 909 (454 1s) after another round I get 0, 0, 0, 0, 0, ...0, 909 (227 0s). It is easy to see now that 909 will be the last number remaining, and since 909 is odd, the answer is to the question is: odd. But is there another way to do this? 


Thank you,

Tina Jin

 
WangDr. Kevin的头像
Re: MCIII Number Theory 2.30 and 2.32
WangDr. Kevin - 2023年12月29日 Friday 22:22
 

For 2.30:

It is a good step to set variables for each number in the $3\times 3$ square.  Let's not sum all the equations, instead add the equations related to the middle row, the middle column, and the two diagonals.

For 2.32:

Many of the questions in this chapter are about parity.  There is one very common and important fact about parity: if $a$ and $b$ are two integers, then $a-b$ and $a+b$ have the same parity (both are even or both are odd).  To see it is true, notice that the difference between these two numbers is $2b$, an even number.

For this particular problem, every time we replace two numbers with their difference, the new number has the same parity as the sum of the two numbers.  Since we only care about the parity of the result (even or odd), then it is the same result if we add the two numbers at every step instead of subtract.

JinTina的头像
Re: MCIII Number Theory 2.30 and 2.32
JinTina - 2024年01月5日 Friday 13:40
 

For 2.30:

I still need some help on this problem. I don't really understand the hint (more so I don't understand how it will help me solve the problem) Here's what I've done so far:

adding up the middle column, middle row, and both diagonals it yeilds

(a+b+c+d+e+f+g+h+i)+3e=4*1999 then I tried some modular arithmetic that didn't help me.

For 2.32:

I think I found a solution based on your hint, basically the sum of the numbers is odd, so the last number is odd. (this is because the sum of any two numbers has the same parity as the difference of those two numbers). 

Thank you,

Tina Jin

WangDr. Kevin的头像
Re: MCIII Number Theory 2.30 and 2.32
WangDr. Kevin - 2024年01月6日 Saturday 00:30
 

2.30: You already got it!  Think about what is $(a+b+c+d+e+f+g+h+i)$?

2.32 is correct.

JinTina的头像
Re: MCIII Number Theory 2.30 and 2.32
JinTina - 2024年01月14日 Sunday 10:18
 

Oh! a+b+c+d+e+f+g+h+i is 3*magic sum=3*1999=0 mod (3), but I proved earlier that a+b+c+d+e+f+g+h+i=1 (mod 3), so contradiction and 1999 cannot be the magic sum. (In fact, only integers, positive and negative, that are congruent to 0 mod 3 can be the magic sum)

Thanks!

Tina Jin