Puzzle for May 8, 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.
Scratchpad
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Hint #1
Add E and A to each side of eq.3: D – E + E + A = F – A + E + A which becomes eq.3a) D + A = F + E Add A and F to each side of eq.4: E – A + A + F = C – F + A + F which becomes E + F = C + A which may be written as eq.4a) F + E = C + A
Hint #2
In eq.4a, replace F + E with D + A (from eq.3a): D + A = C + A Subtract A from both sides of the above equation: D + A – A = C + A – A which makes D = C
Hint #3
In eq.2, replace D with C: B + C = C + F Subtract C from both sides of the above equation: B + C – C = C + F – C which makes B = F
Hint #4
In eq.5, substitute C for D, and F for B: C + C – A = F + E which becomes eq.5a) 2×C – A = F + E
Hint #5
Substitute C + A for F + E (from eq.4a) in eq.5a: 2×C – A = C + A In the above equation, add A to both sides, and subtract C from both sides: 2×C – A + A – C = A + C + A – C which makes C = 2×A and also makes D = C = 2×A
Hint #6
Substitute 2×A for C in eq.4a: F + E = 2×A + A which may be written as eq.4b) E + F = 3×A
Hint #7
Substitute 2×A for C and D, and 3×A for E + F (from eq.4b) in eq.1: A + B + 2×A + 2×A + 3×A = 39 which becomes 8×A + B = 39 Subtract 8×A from both sides of the above equation: 8×A + B – 8×A = 39 – 8×A which becomes eq.1a) B = 39 – 8×A
Hint #8
To make eq.1a true, check several possible values for A and B: If A = 3, then B = 39 – 8×3 = 39 – 24 = 15 If A = 4, then B = 39 – 8×4 = 39 – 32 = 7 If A = 5, then B = 39 – 8×5 = 39 – 40 = –1 If A > 5, then B < –1 If A < 3, then B > 15 Since B must be a one-digit non-negative integer, then B = 7 which makes A = 4 and makes F = B = 7 D = C = 2×A = 2 × 4 = 8
Solution
Substitute 8 for C and D, 4 for A, and 7 for B in eq.5: 8 + 8 – 4 = 7 + E which becomes 12 = 7 + E Subtract 7 from each side of the equation above: 12 – 7 = 7 + E – 7 which makes 5 = E making ABCDEF = 478857