Puzzle for February 29, 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.
* "C ^ D" means "C raised to the power of D".
Scratchpad
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Hint #1
eq.5 may be written as: B – C – D = A – D Add C and D to both sides of the above equation: B – C – D + C + D = A – D + C + D which becomes eq.5a) B = A + C Substitute A + C for B (from eq.5a) in eq.3: A + A + C = C + F Subtract C from both sides: A + A + C – C = C + F – C which simplifies to 2×A = F
Hint #2
In eq.4, replace F with 2×A: 2×A = A + E Subtract A from each side of the above equation: 2×A – A = A + E – A which makes A = E
Hint #3
Substitute A for E, 2×A for F, and A + C for B (from eq.5a) in eq.2: A + 2×A = A + C + C Subtract A from each side: A + 2×A – A = A + C + C – A which becomes 2×A = 2×C Divide both sides by 2: 2×A ÷ 2 = 2×C ÷ 2 which means A = C
Hint #4
Substitute A for C in eq.5a: B = A + A which makes B = 2×A
Hint #5
Substitute A for both C and E in eq.6: A ^ D = A × A which is the same as eq.6a) A ^ D = A ^ 2 To make eq.6a true, either: A = 0 or A = 1 or D = 2
Hint #6
Case 1: Check A = 0 ... If A = 0, then C = E = A = 0 which would make B = F = 2×A = 2×0 = 0 Substituting 0 for A, B, C, E, and F in eq.1 would yield: 0 + 0 + 0 + D + 0 + 0 = 23 which would mean D = 23 Since D must be a one-digit integer, then: D ≠ 23 which means A ≠ 0
Hint #7
Case 2: Check A = 1 ... If A = 1, then C = E = A = 1 which would make B = F = 2×A = 2×1 = 2 Substituting 1 for A and C and E, and 2 for B and F in eq.1 would yield: 1 + 2 + 1 + D + 1 + 2 = 23 which would mean D + 7 = 23 Subtracting 7 from both sides of the above equation would make: D + 7 – 7 = 23 – 7 which would mean D = 16 Since D must be a one-digit integer, then: D ≠ 16 which means A ≠ 1 and therefore makes D = 2
Solution
Substitute A for C and E, 2×A for B and F, and 2 for D in eq.1: A + 2×A + A + 2 + A + 2×A = 23 which simplifies to 7×A + 2 = 23 Subtract 2 from both sides of the above equation: 7×A + 2 – 2 = 23 – 2 which makes 7×A = 21 Divide both sides by 7: 7×A ÷ 7 = 21 ÷ 7 which means A = 3 making C = E = A = 3 B = F = 2×A = 2 × 3 = 6 and ABCDEF = 363236