Puzzle for March 9, 2020  ( )

Scratchpad

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

eq.1) A + B + C + D + E + F = 24 eq.2) C + F = A – F eq.3) B + D = E eq.4) C + D + E = B + F eq.5) E – A = A + D eq.6) B – F = A + F

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

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Hint #1


In eq.5, replace E with B + D (from eq.3): B + D – A = A + D In the equation above, subtract D from each side, and add A to each side: B + D – A – D + A = A + D – D + A which makes eq.5a) B = 2×A


  

Hint #2


In eq.6, replace B with 2×A: 2×A – F = A + F In the above equation, add F to each side, and subtract A from each side: 2×A – F + F – A = A + F + F – A which makes eq.6a) A = 2×F


  

Hint #3


In eq.2, substitute 2×F for A: C + F = 2×F – F which becomes C + F = F Subtract F from both sides of the equation above: C + F – F = F – F which makes C = 0


  

Hint #4


Substitute 0 for C, and B + D for E in eq.4: 0 + D + B + D = B + F which becomes B + 2×D = B + F Subtract B from both sides of the above equation: B + 2×D – B = B + F – B which makes 2×D = F


  

Hint #5


Substitute (2×D) for F in eq.6a: A = 2×(2×D) which makes A = 4×D


  

Hint #6


Substitute (4×D) for A in eq.5a: B = 2×(4×D) which makes B = 8×D


  

Hint #7


Substitute 8×D for B in eq.3: 8×D + D = E which makes 9×D = E


  

Solution

Substitute 4×D for A, 8×D for B, 0 for C, 9×D for E, and 2×D for F in eq.1: 4×D + 8×D + 0 + D + 9×D + 2×D = 24 which simplifies to 24×D = 24 Divide both sides of the above equation by 24: 24×D ÷ 24 = 24 ÷ 24 which means D = 1 making A = 4×D = 4 × 1 = 4 B = 8×D = 8 × 1 = 8 E = 9×D = 9 × 1 = 9 F = 2×D = 2 × 1 = 2 and ABCDEF = 480192