Puzzle for October 23, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
Add both C and E to each side of eq.5: D – E + C + E = A – C + C + E which becomes D + C = A + E which may be written as eq.5a) C + D = A + E In eq.5a, replace C + D with A + B + E (from eq.2): A + B + E = A + E Subtract A and E from both sides of the equation above: A + B + E – A – E = A + E – A – E which makes B = 0
Hint #2
In eq.5a, replace A + E with B + D (from eq.6): C + D = B + D Subtract D from each side of the above equation: C + D – D = B + D – D which makes C = B and means C = B = 0
Hint #3
eq.4 may be written as: F – E – D + F = D – F which becomes 2×F – E – D = D – F Add F and D to both sides of the equation above: 2×F – E – D + F + D = D – F + F + D which becomes eq.4a) 3×F – E = 2×D
Hint #4
Add E to both sides of eq.3: E – A + E = D – E + E which becomes eq.3a) 2×E – A = D In eq.6, substitute 0 for B, and 2×E – A for D (from eq.3a): 0 + 2×E – A = A + E In the above equation, add A to both sides, and subtract E from both sides: 0 + 2×E – A + A – E = A + E + A – E which makes eq.6a) E = 2×A
Hint #5
Substitute (2×A) for E in eq.3a: 2×(2×A) – A = D which becomes 4×A – A = D which makes eq.3b) 3×A = D
Hint #6
Substitute 2×A for E, and (3×A) for D in eq.4a: 3×F – 2×A = 2×(3×A) which becomes 3×F – 2×A = 6×A Add 2×A to both sides of the equation above: 3×F – 2×A + 2×A = 6×A + 2×A which makes 3×F = 8×A Divide both sides by 8: 3×F ÷ 8 = 8×A ÷ 8 which means ⅜×F = A
Hint #7
Substitute (⅜×F) for A in eq.6a: E = 2×(⅜×F) which makes E = ¾×F
Hint #8
Substitute (⅜×F) for A in eq.3b: 3×(⅜×F) = D which makes 1⅛×F = D
Solution
Substitute ⅜×F for A, 0 for B and C, 1⅛×F for D, and ¾×F for E in eq.1: ⅜×F + 0 + 0 + 1⅛×F + ¾×F + F = 26 which simplifies to 3¼×F = 26 Divide both sides of the equation above by 3¼: 3¼×F ÷ 3¼ = 26 ÷ 3¼ which means F = 8 making A = ⅜×F = ⅜ × 8 = 3 D = 1⅛×F = 1⅛ × 8 = 9 E = ¾×F = ¾ × 8 = 6 and ABCDEF = 300968