Puzzle for March 12, 2022 ( )
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 the left and right sides of eq.4 to the left and right sides of eq.6, respectively: A + B + E + B + F = D – E + F + D + E – F which becomes A + 2×B + E + F = 2×D Subtract A and 2×B from both sides of the above equation: A + 2×B + E + F – A – 2×B = 2×D – A – 2×B which becomes eq.6a) E + F = 2×D – A – 2×B
Hint #2
In eq.5, replace E + F with 2×D – A – 2×B (from eq.6a): C + D – B = 2×D – A – 2×B – A which becomes C + D – B = 2×D – 2×A – 2×B In the above equation, subtract D from both sides, and add 2×A and 2×B to both sides: C + D – B – D + 2×A + 2×B = 2×D – 2×A – 2×B – D + 2×A + 2×B which becomes eq.5a) C + B + 2×A = D
Hint #3
In eq.3, substitute C + B + 2×A for D (from eq.5a): C + B + 2×A – A = A – B + C which becomes C + B + A = A – B + C In the above equation, subtract C and A from both sides, and add B to both sides: C + B + A – C – A + B = A – B + C – C – A + B which simplifies to 2×B = 0 which means B = 0
Hint #4
In eq.2, substitute 0 for B: 0 + C = A – 0 which makes C = A
Hint #5
Substitute A for C, and 0 for B in eq.5a: A + 0 + 2×A = D which makes eq.5b) 3×A = D
Hint #6
Substitute (3×A) for D (from eq.5b), and 0 for B in eq.6a: E + F = 2×(3×A) – A – 2×0 which becomes E + F = 6×A – A – 0 which becomes eq.6b) E + F = 5×A
Hint #7
Substitute 0 for B, and 3×A for D (from eq.5b) in eq.4: 0 + F = 3×A + E – F which becomes F = 3×A + E – F In the above equation, subtract 3×A from both sides, and add F to both sides: F – 3×A + F = 3×A + E – F – 3×A + F which becomes eq.4a) 2×F – 3×A = E
Hint #8
Substitute 2×F – 3×A for E (from eq.4a) in eq.6b: 2×F – 3×A + F = 5×A which becomes 3×F – 3×A = 5×A Add 3×A to both sides of the above equation: 3×F – 3×A + 3×A = 5×A + 3×A which makes 3×F = 8×A Divide both sides by 8: 3×F ÷ 8 = 8×A ÷ 8 which makes ⅜×F = A and also makes C = A = ⅜×F
Hint #9
Substitute (⅜×F) for A in eq.5b: 3×(⅜×F) = D which makes 1⅛×F = D
Hint #10
Substitute (⅜×F) for A in eq.4a: 2×F – 3×(⅜×F) = E which becomes 2×F – 1⅛×F = E which makes ⅞×F = E
Solution
Substitute ⅜×F for A and C, 0 for B, 1⅛×F for D, and ⅞×F for E in eq.1: ⅜×F + 0 + ⅜×F + 1⅛×F + ⅞×F + F = 30 which simplifies to 3¾×F = 30 Divide both sides of the above equation by 3¾: 3¾×F ÷ 3¾ = 30 ÷ 3¾ which means F = 8 making A = C = ⅜×F = ⅜ × 8 = 3 D = 1⅛×F = 1⅛ × 8 = 9 E = ⅞×F = ⅞ × 8 = 7 and ABCDEF = 303978