Puzzle for August 1, 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.
* DE is a 2-digit number (not D×E).
** "D ^ B" means "D raised to the power of B".
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Hint #1
In eq.1, replace A + B + C with DE (from eq.4): DE + D + E + F = 20 which may be written as 10×D + E + D + E + F = 20 which becomes 11×D + 2×E + F = 20 Subtract 11×D from both sides of the equation above: 11×D + 2×E + F – 11×D = 20 – 11×D which makes eq.1a) 2×E + F = 20 – 11×D
Hint #2
To make eq.1a true, check several possible values for D, E, and F: If D = 0, then 2×E + F = 20 – 11×0 = 20 – 0 = 20 If D = 1, then 2×E + F = 20 – 11×1 = 20 – 11 = 9 If D = 2, then 2×E + F = 20 – 11×2 = 20 – 22 = –2 If D > 2, then 2×E + F < –2 Since E and F are non-negative integers, then 2×E + F ≥ 0 which means D < 2 making D = 0 or D = 1
Hint #3
Begin checking: D = 0 ... Substituting 0 for D in eq.5 would yield: E ÷ F = 0 ^ B which would become E ÷ F = 0 which would make E = 0
Hint #4
Continue checking: D = 0 ... Substituting 0 for D and E in eq.1a, would yield: 2×0 + F = 20 – 11×0 which becomes 0 + F = 20 – 0 which makes F = 20 Since F must be a one-digit integer, then F ≠ 20 which means D ≠ 0 making D = 1
Hint #5
In eq.5, replace D with 1: E ÷ F = 1 ^ F which becomes E ÷ F = 1 which means E = F
Hint #6
In eq.1a, substitute 1 for D, and E for F: 2×E + E = 20 – 11×1 which becomes 3×E = 20 – 11 which makes 3×E = 9 Divide both sides of the above equation by 3: 3×E ÷ 3 = 9 ÷ 3 which makes E = 3 and also makes F = E = 3
Hint #7
Substitute 1 for D, and 3 for F in eq.3: A = 1 + 3 which makes A = 4
Hint #8
Substitute 1 for D, 4 for A, and 3 for both E and F in eq.2: C + 1 = 4 + 3 + 3 which becomes C + 1 = 10 Subtract 1 from both sides of the equation above: C + 1 – 1 = 10 – 1 which becomes C = 9
Solution
Substitute 4 for A, 9 for C, 1 for D, and 3 for E and F in eq.1: 4 + B + 9 + 1 + 3 + 3 = 20 which simplifies to B + 20 = 20 Subtract 20 from both sides of the above equation: B + 20 – 20 = 20 – 20 which means B = 0 making ABCDEF = 409133