Puzzle for June 5, 2020  ( )

Scratchpad

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

eq.1) A + B + C + D + E + F = 36 eq.2) C + F = D + E eq.3) C – A = D – E eq.4) F = A – D eq.5) B + D – E = E + F eq.6) A – C = D – E – F

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

Scratchpad

 

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Hint #1


Add D to both sides of eq.4: F + D = A – D + D which becomes eq.4a) F + D = A   In eq.3, substitute (F + D) for A (from eq.4a): C – (F + D) = D – E which becomes C – F – D = D – E Add D to both sides of the above equation: C – F – D + D = D – E + D which becomes eq.3a) C – F = 2×D – E


  

Hint #2


Add the left and right sides of eq.2 to the left and right sides of eq.3a, respectively: C – F + (C + F) = 2×D – E + (D + E) which becomes 2×C = 3×D Divide both sides of the above equation by 2: 2×C ÷ 2 = 3×D ÷ 2 which makes eq.2a) C = 1½×D


  

Hint #3


Substitute 1½×D for C in eq.2: 1½×D + F = D + E Subtract D from each side of the equation above: 1½×D + F – D = D + E – D which becomes eq.2b) ½×D + F = E


  

Hint #4


Add E to both sides of eq.5: B + D – E + E = E + F + E which becomes B + D = 2×E + F Substitute (½×D + F) for E (from eq.2b) in the above equation: B + D = 2×(½×D + F) + F which becomes B + D = D + 2×F + F which becomes B + D = D + 3×F Subtract D from each side: B + D – D = D + 3×F – D which makes B = 3×F


  

Hint #5


Substitute F + D for A (from eq.4a), 1½×D for C, and (½×D + F) for E (from eq.2b) in eq.6: F + D – 1½×D = D – (½×D + F) – F which becomes F – ½×D = D – ½×D – F – F which becomes F – ½×D = ½×D – 2×F Add ½×D and 2×F to each side of the equation above: F – ½×D + ½×D + 2×F = ½×D – 2×F + ½×D + 2×F which makes 3×F = D


  

Hint #6


Substitute (3×F) for D in eq.2a: C = 1½×(3×F) which makes C = 4½×F


  

Hint #7


Substitute (3×F) for D in eq.2b: ½×(3×F) + F = E which becomes 1½×F + F = E which makes 2½×F = E


  

Hint #8


Substitute 3×F for D in eq.4a: F + 3×F = A which makes 4×F = A


  

Solution

Substitute 4×F for A, 3×F for B and D, 4½×F for C, and 2½×F for E in eq.1: 4×F + 3×F + 4½×F + 3×F + 2½×F + F = 36 which simplifies to 18×F = 36 Divide both sides of the above equation by 18: 18×F ÷ 18 = 36 ÷ 18 which means F = 2 making A = 4×F = 4 × 2 = 8 B = D = 3×F = 3 × 2 = 6 C = 4½×F = 4½ × 2 = 9 E = 2½×F = 2½ × 2 = 5 and ABCDEF = 869652