Puzzle for February 13, 2021 ( )
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 + E with A + C (from eq.2): A + B = A + C + F Subtract A from both sides of the equation above: A + B – A = A + C + F – A which becomes eq.3a) B = C + F
Hint #2
In eq.4, replace B with C + F (from eq.3a): C + F = C + E Subtract C from both sides of the above equation: C + F – C = C + E – C which makes F = E
Hint #3
In eq.2, substitute F for E: eq.2a) D + F = A + C
Hint #4
Substitute D + F for A + C (from eq.2a) in eq.6: B + F = D + F + D which becomes B + F = 2×D + F Subtract F from each side of the above equation: B + F – F = 2×D + F – F which makes B = 2×D
Hint #5
Substitute 2×D for B, and E for F in eq.3: A + 2×D = D + E + E which becomes A + 2×D = D + 2×E Subtract D from each side of the equation above: A + 2×D – D = D + 2×E – D which becomes eq.3b) A + D = 2×E
Hint #6
Add D to both sides of eq.5: C + D + D = A – D + D which becomes eq.5a) C + 2×D = A Substitute C + 2×D for A (from eq.5a) in eq.3b: C + 2×D + D = 2×E which becomes C + 3×D = 2×E Subtract 3×D from both sides of the equation above: C + 3×D – 3×D = 2×E – 3×D which becomes eq.3c) C = 2×E – 3×D
Hint #7
Substitute 2×D for B, and 2×E – 3×D for C (from eq.3c) in eq.4: 2×D = 2×E – 3×D + E which becomes 2×D = 3×E – 3×D Add 3×D to both sides of the above equation: 2×D + 3×D = 3×E – 3×D + 3×D which makes 5×D = 3×E Divide both sides by 3: 5×D ÷ 3 = 3×E ÷ 3 which makes 1⅔×D = E and also makes F = E = 1⅔×D
Hint #8
Substitute (1⅔×D) for E in eq.3c: C = 2×(1⅔×D) – 3×D which becomes C = 3⅓×D – 3×D which makes C = ⅓×D
Hint #9
Substitute ⅓×D for C in eq.5a: ⅓×D + 2×D = A which makes 2⅓×D = A
Solution
Substitute 2⅓×D for A, 2×D for B, ⅓×D for C, and 1⅔×D for E and F in eq.1: 2⅓×D + 2×D + ⅓×D + D + 1⅔×D + 1⅔×D = 27 which simplifies to 9×D = 27 Divide both sides of the above equation by 9: 9×D ÷ 9 = 27 ÷ 9 which means D = 3 making A = 2⅓×D = 2⅓ × 3 = 7 B = 2×D = 2 × 3 = 6 C = ⅓×D = ⅓ × 3 = 1 E = F = 1⅔×D = 1⅔ × 3 = 5 and ABCDEF = 761355