Puzzle for April 11, 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 positive integer.
* EF is a 2-digit number (not E×F).
Scratchpad
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Hint #1
In eq.3, replace D with A – E (from eq.2): B + A – E = A + E Subtract A from both sides of the above equation: B + A – E – A = A + E – A which becomes B – E = E Add E to both sides: B – E + E = E + E which makes B = 2×E
Hint #2
eq.5 may be written as: 10×E + F = B×B In the above equation, replace B with 2×E: 10×E + F = 2×E × 2×E which becomes 10×E + F = 4×E×E Replace E×E with C (from eq.6): eq.5a) 10×E + F = 4×C
Hint #3
Subtract the left and right sides of eq.4 from the left and right sides of eq.3, respectively: B + D – (C + D) = A + E – (A + F) which is equivalent to B + D – C – D = A + E – A – F which becomes B – C = E – F Multiply both sides of the above equation by 4: 4×(B – C) = 4×(E – F) which may be written as eq.3a) 4×B – 4×C = 4×E – 4×F
Hint #4
Substitute 2×E for B, and (10×E + F) for 4×C (from eq.5a) in eq.3a: 4×2×E – (10×E + F) = 4×E – 4×F which is equivalent to 8×E – 10×E – F = 4×E – 4×F which becomes –2×E – F = 4×E – 4×F Add both 2×E and 4×F to each side of the above equation: –2×E – F + 2×E + 4×F = 4×E – 4×F + 2×E + 4×F which simplifies to 3×F = 6×E Divide both sides by 3: 3×F ÷ 3 = 6×E ÷ 3 which means F = 2×E
Hint #5
In eq.2, add E to each side, and subtract D from each side: D + E – D = A – E + E – D which becomes eq.3b) E = A – D
Hint #6
Subtract both D and F from each side of eq.4: C + D – D – F = A + F – D – F which becomes C – F = A – D Substitute 2×E for F, and E for A – D (from eq.3b) in the above equation: C – 2×E = E Add 2×E to both sides: C – 2×E + 2×E = E + 2×E which becomes eq.4a) C = 3×E
Hint #7
Substitute E×E for C (from eq.6) in eq.4a: E×E = 3×E Divide both sides by E: E×E ÷ E = 3×E ÷ E which means E = 3 making C = 3×E = 3 × 3 = 9 B = F = 2×E = 2 × 3 = 6
Solution
Substitute 3 for E in eq.2: eq.2a) D = A – 3 Substitute 6 for B and F, 9 for C, A – 3 for D (from eq.2a), and 3 for E in eq.1: A + 6 + 9 + A – 3 + 3 + 6 = 37 which simplifies to 2×A + 21 = 37 Subtract 21 from both sides: 2×A + 21 – 21 = 37 – 21 which means 2×A = 16 Divide both sides by 2: 2×A ÷ 2 = 16 ÷ 2 which makes A = 8 making D = A – 3 = 8 – 3 = 5 (from eq.2a) and ABCDEF = 869536