My first cryptogram

Logic Level 1

A B C A B C + A B C C C C \large{\begin{array}{c}&& \color{#3D99F6}A & \color{#D61F06}B & \color{#69047E}C \\ & \color{#3D99F6}A & \color{#D61F06}B & \color{#69047E}C \\ + & \color{#3D99F6}A & \color{#D61F06}B & \color{#69047E}C \\ \hline &\color{#69047E}C & \color{#69047E}C & \color{#69047E}C \end{array}}

A B C = ? \large \overline{\color{#3D99F6}A\color{#D61F06}B \color{#69047E}C} = \ ?


The answer is 185.

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7 solutions

Mohamed Tarek
Nov 29, 2015

Only solution where C+C+C=C is possible is if C=5.

Since the final answer is CCC i.e., 555, and they are 3 equal numbers which add to it, then just divide 555 by 3:

555÷3=185

I don't see why A=B=C=0 is not a possible solution.

Julie Foster - 5 years, 6 months ago

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I know it's not stated in the question, but if A=B=C=0 then what's the point of writing A B C + A B C + A B C = C C C. You can just rewrite all with C C C or A A A or B B B.

Razif FA - 5 years, 6 months ago

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The real question is "why not?". Perhaps this question was meant to encompass this additional bit of complexity. There's essentially no problem in having A=B=C=0, even if you can write it in a simpler way.

Julie Foster - 5 years, 6 months ago

@Julie Foster In a cryptogram,by definition,all the letters stand for distinct digits

Abdur Rehman Zahid - 5 years, 6 months ago

I think this is the best solution. :)

Ian Lamigo - 5 years, 6 months ago

This explanation is more conclusive from result obtained than verification with careful steps. Nevertheless, a half guess is also a good way to answer questions in Brilliant.

Lu Chee Ket - 5 years, 6 months ago
Kay Xspre
Nov 22, 2015

3 ( 100 a + 10 b + c ) = 111 c 3(100a+10b+c) = 111c , or simply 10 a + b = 18 5 c 10a+b=\frac{18}{5}c . Here we will notice that between 1 to 9, the only possible solution is that ( a , b , c ) = ( 1 , 8 , 5 ) (a,b,c) = (1,8,5) , thus, a b c = 185. \overline{abc} = 185.

Harish Sasikumar
Nov 24, 2015
  • Finding C

3C =XC : Only solution is, C=5 ( 3 × 5 = 15 3\times5 = 15 )

Reminder, X = 1

  • Finding B

3B+1(reminder) = X5

i.e 3B = X4 Only solution is, B 8 ( 3 × 8 = 24 3\times8 = 24 )

Reminder, X = 2

  • Finding C

3A+2(reminder) = 5

i.e 3C = 3. Only solution is, A=1

Hence A B C = 185. \overline{ABC} = 185.

Lu Chee Ket
Dec 7, 2015

As A = B = C = 0 is not where we should cease particularly when we know that it is usual to have different digits being meant by different alphabets' representation, we should look for others.

Among 1 to 9, only 5 added itself consecutively for twice such as multiplied by 3 which shall produce digit at 1 0 0 10^0 of its own, therefore, C = 5.

With C = 5, it carries forward a 1 to add onto another digit multiplied by 3. Only 8 × \times 3 + 1 shall give 1 0 1 10^1 of 5 among 0 to 9, therefore, B = 8.

With C = 5 and B = 8, it carries forward a 2 onto another digit (different) multiplied by 3. Only 1 × \times 3 + 2 give 1 0 2 10^2 of 5 among 1 to 9 (first digit is usually meant for non-zero), therefore, A = 1.

850 × \times 3 = 2550 is obviously not the answer, so, 1 0 0 10^0 or C not considered for 0 is correct.

Therefore, 185 is the only answer.

Answer: 185 \boxed{185}

Jingyang Tan
Dec 1, 2015

i copied from the internet

Fidel R.
Nov 30, 2015

Assume A, B, and C are single digit integers. The highest number one of these can be is 9. From the first column: 3C = n(10)+C. Because 3(9)=27, n can't be greater than 2. Therefore, there are three options: 3 C = C 3C=C 3 C = 10 + C 3C=10+C 3 C = 20 + C 3C=20+C Solving each, one gets: C = 0 C=0 C = 5 C=5 C = 10 C=10 5 is the only single digit; therefore, C=5. The second column now is evaluating 3B+1 because you have to "carry the one" you get by getting 15 after adding 5+5+5. By that logic: 3 B + 1 = 5 3B+1=5
3 B + 1 = 15 3B+1=15
3 B + 1 = 25 3B+1=25 Evaluating each, one gets: B = 4 / 3 B=4/3
B = 14 / 3 B=14/3 B = 8 B=8 B is the only integer; therefore, B=8. The sum of the last column has to be exactly 5 because there aren't any more columns after that where you can "carry the one" to. By the same logic as for the second column, the last column is evaluating 2+3A. Therefore: 2 + 3 A = 5 2+3A=5 A = 1 A=1 If C=5, B=8 and A=1, then ABC=185.


Sadasiva Panicker
Nov 30, 2015

185 + 185 + 185 = 555; then ABC = 185

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