X Marks The Square

Number Theory Level 1

Place integers from 1 to 9 (once each) into the 9 squares above, such that the horizontal row sums to 30 and the vertical column sums to 22.

What number would you place in square X?

7 5 3 2

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Normando Mendonça by Normando Mendonça , 67, Brazil

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14 solutions

1 to 9 add up to 45. So if you have 30 in the top row, the numbers in the column below the row add to 15. This means that X must be 7 as 15+7=22

31 Helpful 0 Interesting 0 Brilliant 0 Confused

Sum of Arithmetic progression of first 9 natural numbers/integers is 45 i.e. (1+9)/2 *9=45 Since the sum of the top line is given as 30,the sum mof the remaining four numbersa has to be 45-30=15 That makes the common number at the mid point as 7

Chellappanpillai S. Radhakrishnan - 6 years, 11 months ago

Good answer

Anshuman Singh - 6 years, 11 months ago

Amazing answer.thanks.

Minoo Moradi - 6 years, 11 months ago

how the 15 come.... Can u explain? in my thought u just subtract 30 from 45 or u just half of the 30, i'm really confused...!!!

Libra Khan - 6 years, 11 months ago

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Subtract 30 from 45.

Vaibhav Shete - 6 years, 11 months ago

Please read explanation above.

Chellappanpillai S. Radhakrishnan - 6 years, 11 months ago

Impressive answer, I cheated by using excel and assort the numbers until I found that it's 7.

Anh Vu - 6 years, 11 months ago
Felipe Magalhães
Jul 11, 2014

Calling each square a letter, we can say:

a + b + x + c + d = 30

x + e + f + g + h = 22

Just adding the equations we have...

2x + a + b + c + d + e + f + g + h = 52

So if we sum all the numbers from 1 to 9, but with a number repeated, we should get 52.

On a easier way, we just need to discover the sum from 1 to 9, them make 52 minus it:

x + (x + a + b + c + d + e + f + g + h) = 52 (what is in ( ) are numbers from 1 to 9, not repeated)

x + (1+2+3+4+5+6+7+8+9) = 52

x + 45 = 52

x = 52 - 45

x = 7

And that's our answer! :D

8 Helpful 0 Interesting 0 Brilliant 0 Confused

That is another valid approach, but more tedious.

Chellappanpillai S. Radhakrishnan - 6 years, 11 months ago

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Actually, I believe it is a lot less tedious than your solution. You can think about it this way: only x x is double counted. So if we sum 22 22 and 30 30 it should surpass 1 + 2 + ⋯ + 8 + 9 = 45 1+2+\cdots +8+9=45 by x x . So the answer is just 22 + 30 − 45 = 7 22+30-45=\boxed{7} .

Daniel Liu - 6 years, 11 months ago
William Chau
Jul 24, 2014

x+1+2+...+9 = 30+22,

x+9*10/2 = 52,

x = 7.

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Nıkhıl Lalwani
Jul 13, 2014

1+2+3+4+5+6+7+8+9=45
30 (SUM OF ROW) + 22 (SUM OF COLUMN) = 52
AND
52-45=7 (ANSWER)


1 Helpful 0 Interesting 0 Brilliant 0 Confused
Christian Barrera
Jul 11, 2014

If we add the column and row totals it should be 52 (30+22) All numbers are used once except x which is used twice, for both row and column sum. Given that, we could sum all numbers (1-9 and x) and it should total 52 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + x = 52 45 + x = 52 x = 7

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Aakash Sidhwani
Jul 11, 2014

the sum of no. from 1 to 9 is 45 and if u add the horizonal and vertical row u get 52. the common no. is x which 52-45=7

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Syed Ahsanuddin
Jul 10, 2014

Its so simple if you are familiar with these type of problems. Since 'x' is placed as a common one to both vertical and horizontal ones, and the resulting number given is also considerably high, it can only contain one of the largest numbers(numbers larger than 6), that is 7/8/9. Since 7 is the only option which is larger than 6, the answer must be 7

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Richard Levine
Jul 10, 2014

Let y be the sum of the 4 numbers below x. (1+2+3+4+5+6+7+8+9) - y = 30; 22 - y = x; Now subtract the second equation from the first; 23 = 30 - x; x = 7

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I don't really get what you say.could you explain more?

Minoo Moradi - 6 years, 11 months ago
Venu Gvgk
Jul 10, 2014

The sum of numbers 1 to 9 is 45. In the given arrangement, the number X occurs twice. So, the sum of all the numbers in the given arrangement would be (the sum of 1 to 9) + X. This sum is given to be 30+22 = 52. Hence, X = 7.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

brilliant! i did it by trial and error (quickly, but still!)

Barry Evans - 6 years, 11 months ago

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Trial and error was the first way to learn everything, but once we learn methods for working out such problems, it is better to refresh memory and use the methods instead of repeating the unnecessary method of trial and error.

Chellappanpillai S. Radhakrishnan - 6 years, 11 months ago
Cruel Jupiter
Jul 10, 2014

im guessing my answer. :)

1 Helpful 0 Interesting 0 Brilliant 0 Confused

horizontal: 2,3,7,8,9 vertical: 7,6,5,4,1

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Wow youre a genius

Cyd Redelosa - 6 years, 11 months ago

2+3+7+8+9= 29 not equal to 30 7+6+5+4+1=23 not equal to 22 can you explain you ans?

Uzair Farooqi - 6 years, 11 months ago

check again samantha...

aroop kundu - 6 years, 11 months ago

This is a wrong answert.

Chellappanpillai S. Radhakrishnan - 6 years, 11 months ago

fyou cyd. fyouuu

Samantha Andrea Respecia - 6 years, 11 months ago

only two sets of solutions possible 1. horizontal 9 8 7 4 2 , vertical 7 6 5 3 1 2. horizontal 9 8 7 1 5 vertical 7 6 4 3 2

aroop kundu - 6 years, 11 months ago

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35679 and 45678 also work

Sophie Crane - 6 years, 11 months ago

Its so simple if you are familiar with these type of problems.

Since 'x' is placed as a common one to both vertical and horizontal ones, and the resulting number given is also considerably high, it can only contain one of the largest numbers(numbers larger than 6), that is 7/8/9. Since 7 is the only option which is larger than 6, the answer must be 7

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Summing up numbers to 30, I have 9+8+7+5+1 in horizontal row. Numbers left now, are 6,2,3,4, that has sum of 15; still one number left in vertical row to have a sum of 22. Notice that 15+7 =22, thus, 7 is the common number of horizontal and vertical rows (which is the square where the X was placed).

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The top row can also be 9,8,7,4.2 and still be right

Chellappanpillai S. Radhakrishnan - 6 years, 11 months ago
Krishna Garg
Jul 10, 2014

We know that sum of all numbers from 1 to 9 is 45.Sice horizotal row should be 30 and vertical 22,differece is 8(30-22=8).thereforewe putnumbers 4,5,7(forX)6 and 8 in horizomtal row and remaining numbers i vertical column to satisfy the requitrements.Here for X wer get 7 as Answer.

K.K.GARG,India

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