So Many Possibilities

Algebra Level 2

□ + □ = □ □ + □ = □ □ + □ = □ \large \begin{array} { ccccccc} \square + \square = \square \\ \square + \square = \square \\ \square + \square = \square \\ \end{array}

Can we fill in each of the squares with a distinct digit such that all of the three equations are true?

Yes, we can No, we cannot

Chung Kevin by Chung Kevin , 45, Australia

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2 solutions

Jon Haussmann
Sep 25, 2017

Clearly, we cannot use the digit 0, so we must use all the digits from 1 to 9.

Suppose that such a placement was possible: a + b = c d + e = f g + h = i \begin{aligned} a + b &= c \\ d + e &= f \\ g + h &= i \end{aligned}

Then a + b + c + d + e + f + g + h + i = c + c + f + f + i + i = 2 ( c + f + i ) . a + b + c + d + e + f + g + h + i = c + c + f + f + i + i = 2(c + f + i). But a + b + c + d + e + f + g + h + i = 1 + 2 + ⋯ + 9 = 45 , a + b + c + d + e + f + g + h + i = 1 + 2 + \dots + 9 = 45, which is not even. So such a placement is not possible.

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I was wondering how many solutions would forget about the digit 0. Glad you caught it.

I just posted a slightly harder version .

Chung Kevin - 3 years, 8 months ago
Samir Betmouni
Oct 7, 2017

Since we can't use the zero, we have to use all five odd digits. But each equation needs either 0 or 2 odd numbers. (Since odd+odd=even, etc). But five gives two pairs and one left over.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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