Puzzle for August 29, 2019 ( )
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.
* "A ^ E" means "A raised to the power of E".
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Hint #1
Add B, D, and F to both sides of eq.6: C – D – F + B + D + F = A – B + E + B + D + F which becomes C + B = A + E + D + F which also may be written as eq.6a) B + C = A + D + E + F
Hint #2
In eq.6a, replace B + C with A + F (from eq.5): A + F = A + D + E + F Subtract A and F from both sides of the above equation: A + F – A – F = A + D + E + F – A – F which becomes 0 = D + E Since D and E are non–negative integers, the equation above makes D = 0 and E = 0
Hint #3
Substitute 0 for E in eq.2: A = C + 0 which becomes A = C
Hint #4
Substitute 0 for D in eq.3: 0 = B – F Add F to both sides of the equation above: 0 + F = B – F + F which becomes F = B
Hint #5
Substitute B for F, and 0 for D and E in eq.4: B – C + 0 + B = A ^ 0 which becomes 2×B – C = 1 (assumes A ≠ 0) Add C to both sides: 2×B – C + C = 1 + C which makes eq.4a) 2×B = 1 + C
Hint #6
eq.1 may be written as: B + C + A + D + E + F = 16 Replace A + D + E + F with B + C (from eq.6a): B + C + B + C = 16 which is the same as eq.1a) 2×B + 2×C = 16
Hint #7
In eq.1a, replace 2×B with 1 + C (from eq.4a): 1 + C + 2×C = 16 Subtract 1 from each side: 1 + C + 2×C – 1 = 16 – 1 which makes 3×C = 15 Divide both sides by 3: 3×C ÷ 3 = 15 ÷ 3 which makes C = 5 and which also makes A = C = 5
Solution
Substitute 5 for C in eq.4a: 2×B = 1 + 5 which makes 2×B = 6 Divide both sides by 2: 2×B ÷ 2 = 6 ÷ 2 which means B = 3 which also means F = B = 3 making ABCDEF = 535003