Puzzle for August 8, 2020  ( )

Scratchpad

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

eq.1) A + B + C + D + E + F = 23 eq.2) C + F = D eq.3) A = C + E eq.4) D + F = A + B + C eq.5) A + E – F = C – E eq.6)* CD – E + F = AB

A, B, C, D, E, and F each represent a one-digit non-negative integer.
*  AB and CD are 2-digit numbers (not A×B or C×D).

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Hint #1


Subtract the left and right sides of eq.3 from from the left and right sides of eq.5, respectively: A + E – F – A = C – E – (C + E) which becomes E – F = C – E – C – E which becomes E – F = –2×E Add 2×E and F to both sides of the above equation: E – F + 2×E + F = –2×E + 2×E + F which makes 3×E = F


  

Hint #2


In eq.2, replace F with 3×E: eq.2a) C + 3×E = D


  

Hint #3


In eq.4, replace D with C + 3×E (from eq.2a), F with 3×E, and A with C + E (from eq.3): C + 3×E + 3×E = C + E + B + C which becomes C + 6×E = 2×C + E + B Subtract 2×C and E from both sides of the equation above: C + 6×E – 2×C – E = 2×C + E + B – 2×C – E which becomes eq.4a) 5×E – C = B


  

Hint #4


eq.6 may be written as: 10×C + D – E + F = 10×A + B Substitute C + 3×E for D (from eq.2a), 3×E for F, (C + E) for A (from eq.3), and 5×E – C for B (from eq.4a) in the above equation: 10×C + C + 3×E – E + 3×E = 10×(C + E) + 5×E – C which becomes 11×C + 5×E = 10×C + 10×E + 5×E – C which becomes 11×C + 5×E = 9×C + 15×E Subtract 5×E and 9×C from both sides: 11×C + 5×E – 5×E – 9×C = 9×C + 15×E – 5×E – 9×C which makes 2×C = 10×E Divide each side by 2: 2×C ÷ 2 = 10×E ÷ 2 which makes C = 5×E


  

Hint #5


Substitute 5×E for C in eq.3: A = 5×E + E which makes A = 6×E


  

Hint #6


Substitute 5×E for C in eq.4a: 5×E – 5×E = B which makes 0 = B


  

Hint #7


Substitute 5×E for C in eq.2a: 5×E + 3×E = D which makes 8×E = D


  

Solution

Substitute 6×E for A, 0 for B, 5×E for C, 8×E for D, and 3×E for F in eq.1: 6×E + 0 + 5×E + 8×E + E + 3×E = 23 which simplifies to 23×E = 23 Divide both sides of the equation above by 23: 23×E ÷ 23 = 23 ÷ 23 which means E = 1 making A = 6×E = 6 × 1 = 6 C = 5×E = 5 × 1 = 5 D = 8×E = 8 × 1 = 8 F = 3×E = 3 × 1 = 3 and ABCDEF = 605813