Puzzle for January 25, 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.
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Hint #1
Add A to both sides of eq.5: C – A + A = D + A which becomes C = D + A which is the same as C = A + D In eq.3, replace A + D with C: eq.3a) B + F = C
Hint #2
In eq.4, substitute (B + F) for C (from eq.3a): F – (B + F) = B which is equivalent to F – B – F = B which means –B = B Add B to both sides of the equation above: –B + B = B + B which makes 0 = 2×B which means 0 = B
Hint #3
In eq.3a, substitute 0 for B: 0 + F = C which makes F = C
Hint #4
Substitute 0 for B, and F for C in eq.2: D + E + F = A + 0 + F – D In the above equation, subtract F from both sides, and add D to both sides: D + E + F – F + D = A + 0 + F – D – F + D which becomes eq.2a) 2×D + E = A
Hint #5
Substitute 0 for B, and 2×D + E for A (from eq.2a) in eq.3: 0 + F = 2×D + E + D which makes F = 3×D + E and also makes eq.3b) C = F = 3×D + E
Hint #6
Substitute 2×D + E for A (from eq.2a), 0 for B, and 3×D + E for C and F (from eq.3b) in eq.1: 2×D + E + 0 + 3×D + E + D + E + 3×D + E = 33 which becomes 9×D + 4×E = 33 Subtract 9×D from each side of the above equation: 9×D + 4×E – 9×D = 33 – 9×D which becomes 4×E = 33 – 9×D Divide both sides by 4: 4×E ÷ 4 = (33 – 9×D) ÷ 4 which makes eq.1a) E = (33 – 9×D) ÷ 4
Hint #7
To make eq.1a true, check several possible values for D and E: If D = 0, then E = (33 – 9×0) ÷ 4 = (33 – 0) ÷ 4 = 33 ÷ 4 = 8¼ If D = 1, then E = (33 – 9×1) ÷ 4 = (33 – 9) ÷ 4 = 24 ÷ 4 = 6 If D = 2, then E = (33 – 9×2) ÷ 4 = (33 – 18) ÷ 4 = 15 ÷ 4 = 3¾ If D = 3, then E = (33 – 9×3) ÷ 4 = (33 – 27) ÷ 4 = 6 ÷ 4 = 1½ If D = 4, then E = (33 – 9×4) ÷ 4 = (33 – 36) ÷ 4 = –3 ÷ 4 = –¾ If D > 4, then E < –¾ Since E must be a one-digit non-negative integer, then E = 6 which makes D = 1
Hint #8
Substitute 1 for D, and 6 for E in eq.2a: 2×1 + 6 = A which becomes 2 + 6 = A which means 8 = A
Solution
Substitute 1 for D, and 6 for E in eq.3b: C = F = 3×1 + 6 which becomes C = F = 3 + 6 which means C = F = 9 making ABCDEF = 809169