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

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 A -  3, 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.

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 Dice Date Indicator!

How can you represent days of month using two 6 sided dice? You can write one number on each face of the dice from 0 to 9 and you have to represent days from 1 to 31, for example for 1, one dice should show 0 and another should show 1, similarly for 29 one dice should show 2 and another should show 9.

This is how it can be indicated!

Making of The Dice Date Indicator

What was the challenge?

Dice 1: 0 1 2 3 5 7

Dice 2: 0 1 2 4 6 8

The number 0 has to be present on both the dice. The '0' on first die needed for dates from 1 to 9 to show them as 01,02,03......and the '0' on second die will be used for the dates 10,20,30.

The number 1 and 2 are repeated on the dates 11 and 22 so those 2 numbers has to be there on both dice.

Now, we are left with total 6 positions but 7 numbers - 3 to 9.

However, 6 and 9 can be represented by single die if it is written on one of the side of the die. In normal position it will represent 6 & in inverted position it will show 9 or say vice versa.

For example, the dates 6 and 9 can be indicated as - 

Find Correct Digits For Correct Letters

If each letter represents a different nonzero digit, what must Z be?

Finding difficult? Click Here! 

Correct Digits For Correct Letters

The equation in question was....

First of all X + Y + Z  = Z < = 10 in not possible since in that case, X +  Y + Z - Z =0 i.e. 
X + Y = 0.

Hence, there must be carry 1 forwarded to digit's place. So,

X + Y + Z = 10 + Z

X + Y = 10.          ........(1)

Therefore, possible pairs for (X,Y) or for (Y,X) are (1,9), (2,8), (3,7), (4,6) and (5,5) out of which (5,5) is invalid as repeating digits are not allowed.

Now, since carry is forwarded to ten's place addition, it looks like,

1 + X + Y + Z = XY

Putting (1) in above,

1 + 10 + Z = XY

11 + Z = XY        ........(2)

The maximum value of Z can be 9 & in that case as per above equation XY = 20. But since 0 is not allowed Z must be less than or equal to 8. Then XY <= 11 + 8 = 19.

So X which seems to be carry to hundred place must be 1 (can't be 0 or 2 as proved above). Hence, X = 1.

If X = 1, then from (1), Y = 9.

And if XY = 19 then from (2) Z = 8.

To conclude, X = 1, Y = 9 and Z = 8.

Final equation, looks like,

11 + 99 + 88 = 198.

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Another Method :

The given equation is nothing but

10X + X + 10Y + Y + 10Z + Z = 100X + 10Y + Z

11(X + Y + Z) = 100X + 10Y + Z

89X = Y + 10Z = 10Z + Y

Since, only whole digits are allowed, maximum value of 10Z + Y can be 98 with Z = 9 & Y = 8. So if X = 2 then 89 x 2 = 198 = 10Z + Y is impossible case. Hence, X = 1.

And only Z = 8 and Y = 9 satisfies 89 x 1 = 10Z + Y = 10(8) + 9 = 89.

In short, X = 1, Y = 9 and Z = 8.

The Number Game!

Let’s play a game. You name an integer from 1 to 10. Then we’ll take turns adding an integer from 1 to 10 to the number our opponent has just named, giving the resulting sum as our answer. Whoever reaches 100 first is the winner.

You go first. What number should you choose?

This is how you can be winner!

Winning The Number Game!

What was the game?

Here the player whose number 'forces' sum to fall in range of 90-99 will be ending on losing side. 

That means, somehow if you 'force' the total at some point to 89 then opponent has to fall in the range of 90-99 with his number. 

To get 'door' to total 89 you have to force the previous sum to 78 so that opponent is forced to open a 'door' for total 89 for you.

And so on backward you have to make stops at 67, 56, 45, 34, 23, 12, 1.

So you have to start with 1 & achieve all above milestones.

Let's verify our conclusion. Suppose you started with 1.

You       Sum      Opponent    Sum

1            -             8              9
3           12            5              17
6           23            7              30
4           34            9              43
2           45            4              49
7           56            10            66
1           67            3              70
8           78            2              80
9           89            4              93
7           100           

YOU WIN!

Forgotten Bank Account Number

Today, John has to transfer 50 euro to the bank account of a Dutch friend. He has written down the account number on a piece paper. But since he had forgotten to take out the paper from his trousers when he put them in the washing machine, one digit of the bank account number became unreadable. The note says: 3170?4847. 

The friend of John is climbing the Mount Everest at the moment, so it is impossible for John to call his friend. Suddenly he remembers that a for a valid Dutch bank account number it holds that the first digit times 9 + the second digit times 8 + the third digit times 7 + ...... + the ninth digit times 1 should be divisible by 11. John thinks for a moment and finds the correct number. 

What is it? 

That it is !

Recalling Forgotten Bank Account Number

What were the clues? 

Let x be that missing digit. Then the bank account number looks like, 3170x4847.

Now 9 x First Digit + 8 x Second Digit + 7 x Third Digit.......must be divisible by 11.

That is 27 + 8 + 49 + 0 + 5x + 16 + 24 + 8 + 7 = 139 + 5x must be divisible by 11. 

The only value of x as a digit the satisfies above is 3. With that, total sum 154 is divisible by 11.

Hence, the account number must be, 317034847.