Puzzle for October 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.
* "C ^ E" means "C raised to the power of E".
Scratchpad
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Hint #1
In eq.3, replace A with C + E + F (from eq.2): C + E + F – B = C + F In the above equation, add B to both sides, and subtract C and F from both sides: C + E + F – B + B – C – F = C + F + B – C – F which simplifies to E = B
Hint #2
In eq.4, replace E with B: B – F = B – C Subtract B from both sides of the equation above: B – F – B = B – C – B which becomes –F = –C Multiply both sides by (–1): –F × (–1) = –C × (–1) which means F = C
Hint #3
Substitute B for E, and F for C in eq.5: B + B + F = A – B – F which becomes 2×B + F = A – B – F Add B and F to both sides of the above equation: 2×B + F + B + F = A – B – F + B + F which becomes eq.5a) 3×B + 2×F = A
Hint #4
Substitute 3×B + 2×F for A, and F for C in eq.3: 3×B + 2×F – B = F + F which becomes 2×B + 2×F = 2×F Subtract 2×F from each side of the equation above: 2×B + 2×F – 2×F = 2×F – 2×F which makes 2×B = 0 which means B = 0 and also means E = B = 0
Hint #5
Substitute 0 for B in eq.5a: 3×0 + 2×F = A which makes 2×F = A
Hint #6
Substitute 0 for E in eq.6: D = C ^ 0 which makes D = 1
Solution
Substitute 2×F for A, 0 for B and E, F for C, and 1 for D in eq.1: 2×F + 0 + F + 1 + 0 + F = 17 which becomes 4×F + 1 = 17 Subtract 1 from both sides of the equation above: 4×F + 1 – 1 = 17 – 1 which makes 4×F = 16 Divide both sides by 4: 4×F ÷ 4 = 16 ÷ 4 which means F = 4 making A = 2×F = 2×4 = 8 C = F = 4 and ABCDEF = 804104