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7 Additions and 1 Subtraction

Fill the boxes below with only 7 addition (+) and 1 subtraction (-) operators. 


7 Additions and 1 Subtraction



The smartest way to find the right place 

Correct Placement of Subtraction Operator


This was the question!

For a moment. let's put '+' operator in all given boxes.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 33.

Let 'x' be the number ahead of which '-' should be there where x can be any number from 1 to 9. 

In above addition the x is already (unknown value) added so first we need to subtract it twice from LHS; first to compensate the addition operation & second for the mandatory subtraction operator that is to be placed at right place.

So for example, if it's ahead of 4 as right place then

1 + 2 + 3 + 4 - 4 - 4 + 5 + 6 + 7 + 8 + 9 = 33.

1 + 2 + 3 + (4 - 4) - 4 + 5 + 6 + 7 + 8 + 9 = 33.

For x, which is already there & can be any from 1 to 9,

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9  - x - x = 33. 

45 - 2x = 33

2x = 12

x = 6.

Correct Placement of Subtraction Operator


Hence, the subtraction (-) sign should be ahead of 6

The Third Case of Mystery Number

There is a ten-digit mystery number (no leading 0), represented by ABCDEFGHIJ, where each numeral, 0 through 9, is used once. 

Given the following clues, what is the number?

1) Digit A is either a square number or a triangle number, but not both.


2) Digit B is either an even number or a cube number, but not both.


3) Digit C is either a cube number or a triangle number, but not both.


4) Digit D is either an odd number or a square number, but not both.


5) Digit E is either an odd number or a cube number, but not both.


6) Digit F is either an odd number or a triangle number, but not both.


7) Digit G is either an odd number or a prime number, but not both.


8) Digit H is either an even number or a square number, but not both.


9) Digit I is either a square number or a cube number, but not both.


10) Digit J is either a prime number or a triangle number, but not both.


11) A < B, C < D, E < F, G < H, I < J


12) A + B + C + D + E < F + G + H + I + J


The Third Case of Mystery Number


Here is that MYSTERY number!


The FIRST Case of Mystery Number

The SECOND Case of Mystery Number 

Demystifying The Third Mystery Number


What was the challenge?

Taking a look at the given clues first for identifying the mystery number ABCDEFGHIJ.

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1) Digit A is either a square number or a triangle number, but not both.

2) Digit B is either an even number or a cube number, but not both.


3) Digit C is either a cube number or a triangle number, but not both.


4) Digit D is either an odd number or a square number, but not both.


5) Digit E is either an odd number or a cube number, but not both.


6) Digit F is either an odd number or a triangle number, but not both.


7) Digit G is either an odd number or a prime number, but not both.


8) Digit H is either an even number or a square number, but not both.


9) Digit I is either a square number or a cube number, but not both.


10) Digit J is either a prime number or a triangle number, but not both.


11) A < B, C < D, E < F, G < H, I < J


12) A + B + C + D + E < F + G + H + I + J  


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STEPS  :

Even numbers between 0 to 9 - 0, 2, 4, 6, 8

Odd numbers between 0 to 9 - 1, 3, 5, 7, 9

Square numbers between 0 to 9 - 0, 1, 4, 9 

Cube numbers between 0 to 9 - 8.

Prime numbers between 0 to 9 - 2, 3, 5, 7

Triangle numbers between 0 to 9 - 0, 1, 3, 6

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STEP 1 : 

A can't be 0 as no leading 0's in the mystery number ABCDEFGHIJ.

Remember both of hints can't be true at the same time. That is Digit A is either a square or a triangle number but not both. Since, 0 and 1 are square as well as triangle numbers, A can't be 0 or 1. 

Similarly, 8 is even as well as cube, B can't take 8. And so on.

Possible values of A3, 4, 6, 9. (Square/Triangle)

Possible Values of B - 0, 2, 4, 6. (Even/Cube).

Possible Values of C - 0, 3, 6, 8. (Cube/Triangle)

Possible Values of D - 3, 4, 5, 7. (Odd/Square)

Possible Values of E - 3, 5, 7, 8. (Odd/Cube)

Possible Values of F - 0, 5, 6, 7, 9 (Odd/Triangle)

Possible Values of G - 1, 2, 9 (Odd/Prime)

Possible Values of H - 0, 2, 6, 8, 9 (Even/Square)

Possible Values of I 4, 8, 9 (Square/Cube)

Possible Values of J 0, 1, 2, 5, 6, 7  (Prime/Triangle) 

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STEP 2 :

The value of I can't be 8 or 9 as there will be no digit left for J which is greater than I. 

Hence, I = 4.
 
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STEP 3 : 

Digit A can't be 6 or 9 since no digit will be left for B>A. 

As we have, I = 4, the digit A = 3 and hence B = 6.

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STEP 4 :

The only digit left for C is 0, i.e. C = 0 as C can't be 8 for C<D to be true.

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STEP 5 :

  Since I = 4 itself, so digit J can't be 0, 1, 2 for I<J to be true.

Digit D and digit J take either take 5 or 7. With that the only valid digit left for E is 8 to satisfy the condition E<F. 

Hence E = 8.

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STEP 6 :

If E = 8, then F has to be 9. So, F = 9.
 
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STEP 7 : 

With 6, 8, 9 already taken up by other letters, the only digit left for H is 2 for G<H to be true. So, H = 3 and hence G = 1.

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STEP 8 :

As per condition 12, 

A + B + C + D + E < F + G + H + I + J 

3 + 6 + 0 + D + 8 < 9 + 1 + 2 + 4 + J

17 + D < 16 + J 

As deduced in STEP 4, D must be 5 or 7 and J must be 7 or 5.

For above equation to be true, D = 5 and J = 7.


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Conclusion :

A = 3, B = 6, C = 0, D = 5, E = 8, F = 9, G = 1, H = 2, I = 4, J = 7

Hence, the number ABCDEFGHIJ must be 3605891247.

Demystifying The Third Mystery Number


Verifying the given hints - 

1) 3 is a triangle number, but not a square number.
2) 6 is an even number, but not a cube number.
3) 0 is a cube number, but not a triangle number.
4) 5 is an odd number, but not a square number.
5) 8 is a cube number, but not an odd number.
6) 9 is an odd number, but not a triangle number.
7) 1 is an odd number, but not a prime number.
8) 2 is an even number, but not a square number.
9) 4 is a square number, but not a cube number.
10) 7 is a prime number, but not a triangle number.
11) 3 < 6, 0 < 5, 8 < 9, 1 < 2, 4 < 7
12) 3 + 6 + 0 + 5 + 8 = 22, 9 + 1 + 2 + 4 + 7 = 23, 22 < 23
 
 



The Second Case of Mystery Number

There is a ten-digit mystery number (no leading 0), represented by ABCDEFGHIJ, where each numeral, 0 through 9, is used once. Given the following clues, what is the number?

1) Either A = B / 3 or A = G + 3.


2) Either B = I - 4 or B = E + 4.


3) Either C = J + 2 or C = F * 3.


4) Either D = G * 4 or D = E / 3.


5) Either E = J - 1 or E = D / 4.


6) Either F = B * 2 or F = A - 4.


7) Either G = F + 1 or G = I - 3.


8) Either H = A / 2 or H = C * 3.


9) Either I = H + 3 or I = D / 2.


10) Either J = H - 2 or J = C * 2.


The Second Case of Mystery Number


HERE is that MYSTERY Number! 

What was the FIRST case? 

Demystifying The Second Mystery Number


What was the challenge?

Let's take a look at clues given once again to identify number ABCDEFGHIJ.

------------------------------------------------------------------------------ 


1) Either A = B / 3 or A = G + 3.

2) Either B = I - 4 or B = E + 4.


3) Either C = J + 2 or C = F * 3.


4) Either D = G * 4 or D = E / 3.


5) Either E = J - 1 or E = D / 4.


6) Either F = B * 2 or F = A - 4.


7) Either G = F + 1 or G = I - 3.


8) Either H = A / 2 or H = C * 3.


9) Either I = H + 3 or I = D / 2.


10) Either J = H - 2 or J = C * 2.


------------------------------------------------------------------------------  

STEPS :  

------------------------------------------------------------------------------ 

STEP 1 : 

Few digits can be eliminated for few numbers straightaway on the first go. Those digits just don't 'fit' into the both equations provided for the particular letter.

1) Either A = B / 3 or A = G + 3.

    A - 0.

3) Either C = J + 2 or C = F * 3.

    C - 0, 1.

4) Either D = G * 4 or D = E / 3.


    D - 0, 5, 6, 7, 9

5) Either E = J - 1 or E = D / 4.


    E - 9 

6) Either F = B * 2 or F = A - 4.


    F - 7, 9

7) Either G = F + 1 or G = I - 3.


    G - 8 (since F can't be 7 & I can't be 11) 

8) Either H = A / 2 or H = C * 3.


    H - 0, 5, 7, 8 

9) Either I = H + 3 or I = D / 2.


    I - 8 (since H can't be 5 and obviously D can't be 16).

10) Either J = H - 2 or J = C * 2.


    J - 3 (since H can't be 5 & C can't be 1.5), 5 (since H can't be 7 & C can't 
    be  2.5), 9.

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STEP 2 :

Now, after eliminating some digits for letters we can revise the list of digits a particular letter(s) on LHS of equation can't make because the letter(s) on RHS doesn't (don't) take some digits.

For example, if J can't be 3 or 5 then C = J + 2 can't be 5 or 7. The deduction is supported by the other equation as F can't be 5/3 or 7/3. Hence, C can't be 5 or 7 for sure.

3) Either C = J + 2 or C = F * 3.

    C - 0, 1, 5, 7

4) Either D = G * 4 or D = E / 3.

    D - 0, 5, 6, 7, 9, 3 ( since E can't be 9 & G can't be 9/4).

5) Either E = J - 1 or E = D / 4.

    E - 9, 4 (J can't be 5 & D can't be 16), 8 (J can't be 9 & D can't be 32).

7) Either G = F + 1 or G = I - 3.

    G - 8, 0 [since G can't be -1 and I can't be 3 (since if I = 3 then I = H +3 
     gives H = 0 but H doesn't take 0)].

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STEP 3 : 

For B,

2) Either B = I - 4 or B = E + 4.

    B - 8 (since I can't be 12 & E can't be 4) 

------------------------------------------------------------------------------  

STEP 4 : 

Possible values left for D are -  1, 2, 4, 8. Remember, only one of the two given hints for the particular letter has to be true.
   
   ------------------------------------------------------------------------------

      STEP 4.1 : 

       If D = 1, then D = E/3 (Hint 4) gives E = 3 & hence E = J - 1 (Hint 5) 
      gives J = 4. 

      Here, D = G*4 of Hint 4 must not be true as G would be 1/4.

      Similarly, E = D/4 of Hint 5 must be false and hence other hint i.e. 
      E = J - 1 must be true.

      Following the same logic as above -

      If D = 1 ---> E = 3 ----> E = J - 1 ---> J = 4 ----> 
      ---> J = H - 2 or J = C * 2.

      Hence, H = 6 or C = 2.

      If H = 6, then out of H = A / 2 or H = C * 3 only H = C*3 remains 
      valid which gives C = 2.

      And if C = 2, then C = J + 2 (Hint 3) gives J = 0.

      So, D = 1 produces 2 different values of J as 0 and 4. 

      Hence, this value of D is invalid. 

      ------------------------------------------------------------------------------  

     STEP 4.2 : 

     If D = 2, then (using Hint 4) E = 6 --> (Hint 5)---> J = 7  --->(Hint 10)

     ---> H = 9 ---> makes I = H + 3 invalid hence I = D/2 gives I = 1 

      I = 1 ---> leaves G = F + 1 (of Hint 7) valid.

     Further, H = 9 ---> (Hint 8) ---> C = 3 ---> (Hint 3) ---> 

      ---> F = 1 (since C = J + 2 = 9 invalid as H = 9 already) ---> (Hint 7)

      ---> G = 2.

     So if D = 2, the value of G will be also 2. That's against the rule.

      ------------------------------------------------------------------------------ 

     STEP 4.3 :

     If D = 4, then (using Hint 4) G = 1 ----> (Hint 7) ---> 
     ---> F = 0 or I = 4 (invalid as D = 4 already) ---> (Hint 6) ---> A = 4. 

     Again, if D = 4, then A = 4 also hence this value of D is invalid.

     Hence, the only valid value of D is 8.

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STEP 5 : 

If D = 8, then using Hint 4 we have, G = 2  --->(Hint 7) ---> F = 1 or I =5

If I = 5, --->(Hint 9) ----> H = 2 but already we have G = 2. So,

G = 2  --->(Hint 7) ---> F = 1 ---> (Hint 6) ---> A = 5.

So far, we get,  D = 8, G = 2, F = 1, A = 5 at this stage so far.

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STEP 6 : 

Possible values left for E are - 0, 3, 6 and 7.
 
E = 0 ---> (Hint 5 ) ---> J = 1. But 1 is already taken by F.

E = 3 ---> (Hint 5 ) ---> J = 4 ---> (Hint 10) ---> H = 6 or invalid C = 2

H = 6 ---> (Hint 8 ) ---> invalid C = 2.

The digit 2 is already taken by G, so C = 2 is invalid. Hence, E = 3 is invalid.

E = 7 ---> (Hint 5 ) ---> J = 8. But 8 is already taken by G.

So, the only valid value of E = 6.

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STEP 7 :

E = 6 ---> (Hint 5 ) ---> J = 7 ---> (Hint 10) ---> H = 9 ---> (Hint 5) --->

H = 9 ---> (Hint 8) ---> C = 3

------------------------------------------------------------------------------  

STEP 8 :

The values we got so far,  E = 6, J = 7, H = 9, C = 3, D = 8, G = 2, F = 1, 
A = 5.

Only letters left are B and I while digits left are 0 and 4.

Correct hint out of "Either B = I - 4 or B = E + 4" must be B = I - 4 since 
B = E + 4 = 6 + 4 = 10 must be invalid.

So, B = I - 4 gives us I = 4 and B = 0 .

------------------------------------------------------------------------------  

Conclusion :

A = 5, B = 0, C = 3, D = 8, E = 6, F = 1, G = 2, H = 9, I = 4, J = 7 

all together gives us number

ABCDEFGHIJ as 5038612947.

Demystifying The Second Mystery Number


Let's verify the above number as per given hints.

1) A = G + 3 = 2 + 3 = 5.
2) B = I - 4 = 4 - 4 = 0.
3) C = F * 3 = 1 * 3 = 3.
4) D = G * 4 = 2 * 4 = 8.
5) E = J - 1 = 7 - 1 = 6.
6) F = A - 4 = 5 - 4 = 1.
7) G = F + 1 = 1 + 1 = 2.
8) H = C * 3 = 3 * 3 = 9.
9) I = D / 2 = 8 / 2 = 4.
10) J = H - 2 = 9 - 2 = 7.

 
 
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