Puzzle for June 16, 2023 ( )
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.5, replace F with D + E (from eq.3): C - D = D + E - B Add D and B to both sides of the above equation: C - D + D + B = D + E - B + D + B which becomes eq.5a) C + B = 2×D + E
Hint #2
In eq.5a, replace C with B + E (from eq.2): B + E + B = 2×D + E which becomes 2×B + E = 2×D + E Subtract E from each side of the equation above: 2×B + E - E = 2×D + E - E which makes 2×B = 2×D Divide both sides by 2: 2×B ÷ 2 = 2×D ÷ 2 which makes B = D
Hint #3
In eq.4, substitute B for D: A + B = C + B Subtract B from each side of the above equation: A + B - B = C + B - B which makes A = C
Hint #4
Substitute B for D in eq.5: C - B = F - B Add B to both sides of the above equation: C - B + B = F - B + B which makes C = F and also makes A = C = F
Hint #5
Substitute F for A, and D for B in eq.6: F ÷ D = F × D Multiply both sides of the above equation by D: D × (F ÷ D) = D × F × D which becomes F = D² × F Since D ≠ 0 (from eq.6), then in the above equation, either: F = 0 or D = 1
Hint #6
Begin checking: F = 0 ... If F = 0, then: A = C = F = 0 Substituting 0 for A, C, and F in eq.1 would yield: 0 + B + 0 + D + E + 0 = 35 which would become B + D + E = 35
Hint #7
Finish checking: F = 0 ... Since B, D, and E are one-digit integers, then: B ≤ 9 and D ≤ 9 and E ≤ 9 making B + D + E ≤ 9 + 9 + 9 which means B + D + E ≤ 27 The above inequality means: B + D + E ≠ 35 which means F ≠ 0 and therefore makes D = 1 and also makes B = D = 1
Hint #8
Substitute 1 for D in eq.3: F = 1 + E which also makes eq.3a) A = C = F = 1 + E
Hint #9
Substitute 1 + E for A and C and F (from eq.3a), and 1 for B and D in eq.1: 1 + E + 1 + 1 + E + 1 + E + 1 + E = 33 which simplifies to 5 + 4×E = 33 Subtract 5 from each side of the above equation: 5 + 4×E - 5 = 33 - 5 which makes 4×E = 28 Divide both sides by 4: 4×E ÷ 4 = 28 ÷ 4 which means E = 7
Solution
Substitute 7 for E in eq.3a: A = C = F = 1 + 7 which makes A = C = F = 8 and makes ABCDEF = 818178