Puzzle for February 24, 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
In eq.3, replace D with E + F (from eq.2): E + F + F = C + E which becomes E + 2×F = C + E Subtract E from both sides of the above equation: E + 2×F – E = C + E – E which makes 2×F = C
Hint #2
In eq.6, replace C with 2×F: 2×F + F = A + B – 2×F + D which becomes 3×F = A + B – 2×F + D In the above equation, add 2×F to both sides, and subtract D from both sides: 3×F + 2×F – D = A + B – 2×F + D + 2×F – D which becomes eq.6a) 5×F – D = A + B
Hint #3
In eq.4, substitute 5×F – D for A + B (from eq.6a), and 2×F for C: 5×F – D + E = 2×F + D In the above equation, add D to both sides, and subtract 5×F from both sides: 5×F – D + E + D – 5×F = 2×F + D + D – 5×F which becomes eq.4a) E = 2×D – 3×F
Hint #4
Substitute (E + F) for D (from eq.2) into eq.4a: E = 2×(E + F) – 3×F which is equivalent to E = 2×E + 2×F – 3×F which becomes E = 2×E – F In the equation above, subtract E from both sides, and add F to both sides: E – E + F = 2×E – F – E + F which makes F = E
Hint #5
Substitute F for E in eq.2: F + F = D which makes 2×F = D
Hint #6
In eq.4, substitute F for E, and 2×F for C and D: A + B + F = 2×F + 2×F which becomes A + B + F = 4×F Subtract B and F from each side of the equation above: A + B + F – B – F = 4×F – B – F which becomes eq.4b) A = 3×F – B
Hint #7
Substitute F for E, and (3×F – B) for A (from eq.4b) in eq.5: F – (3×F – B) + B = (3×F – B) – B which is the same as F – 3×F + B + B = 3×F – B – B which becomes –2×F + 2×B = 3×F – 2×B Add 2×F and 2×B to both sides of the equation above: –2×F + 2×B + 2×F + 2×B = 3×F – 2×B + 2×F + 2×B which becomes 4×B = 5×F Divide both sides by 4: 4×B ÷ 4 = 5×F ÷ 4 which makes B = 1¼×F
Hint #8
Substitute 1¼×F for B in eq.4b: A = 3×F – 1¼×F which makes A = 1¾×F
Solution
Substitute 1¾×F for A, 1¼×F for B, 2×F for C and D, and F for E in eq.1: 1¾×F + 1¼×F + 2×F + 2×F + F + F = 36 which simplifies to 9×F = 36 Divide both sides of the above equation by 9: 9×F ÷ 9 = 36 ÷ 9 which means F = 4 making A = 1¾×F = 1¾ × 4 = 7 B = 1¼×F = 1¼ × 4 = 5 C = D = 2×F = 2 × 4 = 8 E = F = 4 and ABCDEF = 758844