Puzzle for March 2, 2020  ( )

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Find the 6-digit number ABCDEF by solving the following equations:

eq.1) A + B + C + D + E + F = 17 eq.2) D + E + F = B + C eq.3) A + E = B + D eq.4) B = D – F eq.5) A – D = B eq.6) C + F = B

A, B, C, D, E, and F each represent a one-digit non-negative integer.

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Hint #1


In eq.3, replace B with A – D (from eq.5): A + E = A – D + D which becomes A + E = A Subtract A from each side of the above equation: A + E – A = A – A which makes E = 0


  

Hint #2


In eq.2, replace E with 0, and replace B with D – F (from eq.4): D + 0 + F = D – F + C which becomes D + F = D – F + C In the above equation, subtract D from both sides, and add F to both sides: D + F – D + F = D – F + C – D + F which makes 2×F = C


  

Hint #3


In eq.6, substitute 2×F for C: 2×F + F = B which makes 3×F = B


  

Hint #4


Substitute 3×F for B in eq.4: 3×F = D – F Add F to each side of the equation above: 3×F + F = D – F + F which makes 4×F = D


  

Hint #5


Substitute 4×F for D in eq.5: A – 4×F = 3×F Add 4×F to both sides of the equation above: A – 4×F + 4×F = 3×F + 4×F which makes A = 7×F


  

Solution

Substitute 7×F for A, 3×F for B, 2×F for C, 4×F for D, and 0 for E in eq.1: 7×F + 3×F + 2×F + 4×F + 0 + F = 17 which simplifies to 17×F = 17 Divide both sides of the above equation by 17: 17×F ÷ 17 = 17 ÷ 17 which means F = 1 making A = 7×F = 7 × 1 = 7 B = 3×F = 3 × 1 = 3 C = 2×F = 2 × 1 = 2 D = 4×F = 4 × 1 = 4 and ABCDEF = 732401