## Feynman’s Puzzle

### June 12, 2009

Since there is no remainder, and it’s easier to work with multiplication than division, we restate the problem: determine *bAc* × *deAf* subject to various constraints.

We begin by establishing variables for *A*, *b*, *c*, *d*, *e* and *f*, ensuring that *A* is different from the others, and calculating *bAc* × *deAf*. The product must be a seven-digit number, and the fifth digit (fourth counting from zero) must be *A*. The first line of the division shows that *d* × *bAc* must be *AA* modulo 100, and the second line of the division shows that *e* × *bAc* must be *A* modulo 10. Finally, from the third line of the division we know that *A* × *bAc* is a four-digit number, and its second digit is *A*. Here is the first attempt at a solution:

`> (list-of (list (list b a c) (list d e a f))`

; initialize digits

(a range 10) (b range 1 10) (c range 10)

(d range 1 10) (e range 10) (f range 10)

; A differs from the other digits

(not (= a b)) (not (= a c)) (not (= a d))

(not (= a e)) (not (= a f))

; calculate the two numbers

(n1 is (+ (* b 100) (* a 10) c))

(n2 is (+ (* d 1000) (* e 100) (* a 10) f))

; answer must be seven digits

(< 999999 (* n1 n2) 10000000)

; fifth digit of answer must be A

(= (list-ref (digits (* n1 n2)) 4) a)

; d * bAc must be AA modulo 100

(= (modulo (* d n1) 100) (+ (* a 10) a))

; e * bAc must be A modulo 10

(= (modulo (* e n1) 10) a)

; A * bAc must be four digits

(< 999 (* a n1))

; second digit of A * bAc must be A

(= (list-ref (digits (* a n1)) 1) a))

(((4 3 7) (9 9 3 9)) ((4 8 4) (7 2 8 9)) ((4 8 4) (7 7 8 9)))

So there are three possibilities, and we have to do some more work. We know that *e* must be less than *d*, since *e* × *bAc* is a three-digit number but *d* × *bAc* is a four-digit number:

`> (list-of (list (list b a c) (list d e a f))`

; initialize digits

(a range 10) (b range 1 10) (c range 10)

(d range 1 10) (e range 10) (f range 10)

; A differs from the other digits

(not (= a b)) (not (= a c)) (not (= a d))

(not (= a e)) (not (= a f))

; calculate the two numbers

(n1 is (+ (* b 100) (* a 10) c))

(n2 is (+ (* d 1000) (* e 100) (* a 10) f))

; answer must be seven digits

(< 999999 (* n1 n2) 10000000)

; fifth digit of answer must be A

(= (list-ref (digits (* n1 n2)) 4) a)

; d * bAc must be AA modulo 100

(= (modulo (* d n1) 100) (+ (* a 10) a))

; e * bAc must be A modulo 10

(= (modulo (* e n1) 10) a)

; A * bAc must be four digits

(< 999 (* a n1))

; second digit of A * bAc must be A

(= (list-ref (digits (* a n1)) 1) a)

; e must be less than d

(< e d))

(((4 8 4) (7 2 8 9)))

This solution uses list comprehensions and the `digits`

function from the Standard Prelude. The completed long division looks like this:

` 7 2 8 9`

- - - - - - - -

4 8 4 ) 3 5 2 7 8 7 6

3 3 8 8

- - - -

1 3 9 8

9 6 8

- - - -

4 3 0 7

3 8 7 2

- - - -

4 3 5 6

4 3 5 6

- - - -

0

This puzzle appeared on Reddit a few weeks ago. Keith Lofstrom tells the story of this puzzle and gives a manual solution at http://wiki.keithl.com/index.cgi?FeynmanPuzzle. You can run the solution given above at http://programmingpraxis.codepad.org/50ygwf6g.

What a fun puzzle!

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