Riddlers for 4 May 2018

Spoilers for last week’s fivethirtyeight.com Riddler Express and Classic:

Express

From Charlie Drinnan, find the letters’ numbers:

If A, B, C, D and E are all unique digits, what values would work with the following equation?

ABC,CDE × 4 = EDC,CBA

No problem. A must be even, and 4A < 10, so A=2. Then 4AB < 100 so B < 5, and BA\mod 4 = 0 so B = 1 and E = 8, or B = 3 and E = 9. But 4E \mod 10 = A = 2 so E = 8.

Now 4D+3 \mod 10 = B = 1 so 4D \mod 10 = 8. D\ne 2 so D = 7.

Finally 4C+3 \mod 10 = C or 3C\mod 10 = 7, so C = 9.

219,978 \times 4= 879,912.

Classic

This is fairly mechanical although I tripped up several times on the way.

Let n_i be the number of coconuts found by the ith pirate, and n_8 be the number of coconuts found in the morning. n_8 is divisible by 7 so write n_8 = 7m_8.

But n_8 is what was left when the seventh pirate discarded one coconut out of n_7 and hid one seventh of the rest, so n_7 = (7/6)n_8+1 = (7/6)(7m_8)+1. For that to be an integer m_8 must be a multiple of 6, so write m_8 = 6m_7. Then n_7 = 7^2m_7+1.

Now n_6 = (7/6)n_7+1 = (7/6)(7^2m_7+1)+1. For integer value, 7^2m_7\mod 6 = m_7\mod 6 = 5, so write m_7 = 6m_6+5. Then n_6 = (7/6)(7^2(6m_6+5)+1)+1 = 7^3m_6+288.

You see how this goes.

n_5 = (7/6)n_6+1 = (7/6)(7^3m_6+288)+1. For integer value, m_6\mod 6 = 0, so write m_6 = 6m_5. Then n_5 = (7/6)(7^3(6m_5)+288))+1 = 7^4m_5+337.

n_4 = (7/6)n_5+1 = (7/6)(7^4m_5+337)+1. For integer value, m_5\mod 6 = 5, so write m_5 = 6m_4+5. Then n_4 = (7/6)(7^4(6m_4+5)+337)+1 = 7^5m_4+14400.

n_3 = (7/6)n_4+1 = (7/6)(7^5m_4+14400)+1. For integer value, m_4\mod 6 = 0, so write m_4 = 6m_3. Then n_3 = (7/6)(7^5(6m_3)+14400)+1 = 7^6m_3+16801.

n_2 = (7/6)n_3+1 = (7/6)(7^6m_3+16801)+1. For integer value, m_6\mod 6 = 5, so write m_3 = 6m_2+5. Then n_2 = (7/6)(7^6(6m_2+5)+16801)+1 = 7^7m_2+705888.

n_1 = (7/6)n_2+1 = (7/6)(7^7m_2+705888)+1. For integer value, m_2\mod 6 = 0, so write m_2 = 6m_1. Then n_1 = (7/6)(7^7(6m_1)+705888)+1 = 7^8m_1+823537. The smallest value corresponds to m_1 = 0 so n_1^{min} = 823537.

These pirates were either very sleepy, or incredibly quick at counting coconuts.

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