Puzzle for October 3, 2020 ( )
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 A and F to both sides of eq.5: F – A + A + F = B + E – F + A + F which becomes 2×F = B + E + A In the above equation, replace B + E with A + D (from eq.4): 2×F = A + D + A which becomes 2×F = 2×A + D Subtract D from each side of the above equation: 2×F – D = 2×A + D – D which becomes eq.5a) 2×F – D = 2×A
Hint #2
Add A to both sides of eq.6: A + B + A = C + D – A + A which becomes eq.6a) 2×A + B = C + D In eq.6a, replace 2×A with 2×F – D (from eq.5a): 2×F – D + B = C + D Add D to both sides of the above equation: 2×F – D + B + D = C + D + D which becomes eq.6b) 2×F + B = C + 2×D
Hint #3
In eq.3, substitute C + E + F for D (from eq.2): C + E + F – B = F – E Subtract F from both sides of the equation above: C + E + F – B – F = F – E – F which becomes C + E – B = –E Add B and E to both sides: C + E – B + B + E = –E + B + E which becomes eq.3a) C + 2×E = B
Hint #4
Substitute C + 2×E for B (from eq.3a) in eq.6b: 2×F + C + 2×E = C + 2×D Subtract C from each side of the above equation: 2×F + C + 2×E – C = C + 2×D – C which becomes 2×F + 2×E = 2×D Divide both sides by 2: (2×F + 2×E) ÷ 2 = 2×D ÷ 2 which becomes eq.6c) F + E = D
Hint #5
Substitute F + E for D (from eq.6c) in eq.2: C + E + F = F + E Subtract E and F from each side of the equation above: C + E + F – E – F = F + E – E – F which simplifies to C = 0
Hint #6
Substitute 0 for C in eq.6a: 2×A + B = 0 + D which becomes eq.6d) 2×A + B = D
Hint #7
Substitute 2×A + B for D (from eq.6d) in eq.4: B + E = A + 2×A + B which becomes B + E = 3×A + B Subtract B from both sides of the equation above: B + E – B = 3×A + B – B which makes E = 3×A
Hint #8
Substitute 0 for C, and (3×A) for E in eq.3a: 0 + 2×(3×A) = B which makes 6×A = B
Hint #9
Substitute 6×A for B, and 0 for C in eq.6d: 2×A + 6×A = D which makes 8×A = D
Hint #10
Substitute 3×A for E, and 8×A for D in eq.6c: F + 3×A = 8×A Subtract 3×A from each side of the equation above: F + 3×A – 3×A = 8×A – 3×A which makes F = 5×A
Solution
Substitute 6×A for B, 0 for C, 8×A for D, 3×A for E, and 5×A for F in eq.1: A + 6×A + 0 + 8×A + 3×A + 5×A = 23 which simplifies to 23×A = 23 Divide both sides of the above equation by 23: 23×A ÷ 23 = 23 ÷ 23 which means A = 1 making B = 6×A = 6 × 1 = 6 D = 8×A = 8 × 1 = 8 E = 3×A = 3 × 1 = 3 F = 5×A = 5 × 1 = 5 and ABCDEF = 160835