Puzzle for March 31, 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 E and C to both sides of eq.4: B – E + E + C = E – C + E + C which becomes eq.4a) B + C = 2×E In eq.2, replace B + C with 2×E (from eq.4a): D = 2×E
Hint #2
In eq.5, replace D with 2×E: F – B = 2×E – F In the above equation, add F and B to both sides, and subtract 2×E from both sides: F – B + F + B – 2×E = 2×E – F + F + B – 2×E which becomes eq.5a) 2×F – 2×E = B
Hint #3
In eq.4a, substitute 2×F – 2×E for B (from eq.5a): 2×F – 2×E + C = 2×E In the above equation, subtract 2×F from both sides, and add 2×E to both sides: 2×F – 2×E + C – 2×F + 2×E = 2×E – 2×F + 2×E which becomes eq.4b) C = 4×E – 2×F
Hint #4
Substitute 4×E – 2×F for C (from eq.4b) in eq.3: E + F = A + 4×E – 2×F In the above equation, subtract 4×E from both sides, and add 2×F to both sides: E + F – 4×E + 2×F = A + 4×E – 2×F – 4×E + 2×F which becomes eq.3a) 3×F – 3×E = A
Hint #5
In eq.6, substitute 2×F – 2×E for B (from eq.5a), 3×F – 3×E for A (from eq.3a), 4×E – 2×F for C (from eq.4b), and 2×E for D: 2×F – 2×E + F = 3×F – 3×E + 4×E – 2×F + 2×E – F which becomes 3×F – 2×E = 3×E Add 2×E to both sides of the above equation: 3×F – 2×E + 2×E = 3×E + 2×E which becomes 3×F = 5×E Divide both sides by 3: 3×F ÷ 3 = 5×E ÷ 3 which makes F = 1⅔×E
Hint #6
Substitute (1⅔×E) for F in eq.3a: 3×(1⅔×E) – 3×E = A which becomes 5×E – 3×E = A which makes 2×E = A
Hint #7
Substitute (1⅔×E) for F in eq.5a: 2×(1⅔×E) – 2×E = B which becomes 3⅓×E – 2×E = B which makes 1⅓×E = B
Hint #8
Substitute (1⅔×E) for F in eq.4b: C = 4×E – 2×(1⅔×E) which becomes C = 4×E – 3⅓×E which makes C = ⅔×E
Solution
Substitute 2×E for A and D, 1⅓×E for B, ⅔×E for C, and 1⅔×E for F in eq.1: 2×E + 1⅓×E + ⅔×E + 2×E + E + 1⅔×E = 26 which simplifies to 8⅔×E = 26 Divide both sides of the above equation by 8⅔: 8⅔×E ÷ 8⅔ = 26 ÷ 8⅔ which means E = 3 making A = D = 2×E = 2 × 3 = 6 B = 1⅓×E = 1⅓ × 3 = 4 C = ⅔×E = ⅔ × 3 = 2 F = 1⅔×E = 1⅔ × 3 = 5 and ABCDEF = 642635