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Sunday, October 19, 2014

A Classic Puzzle


;; A Classic Puzzle

;; This problem, my sources inform me, can be solved by pre-school
;; children in 5-10 minutes, by programmers - in 1 hour, and by
;; mathematicians ... well, check it yourself! :)

(def input (quote ("8809" "7111" "2172" "6666" "1111" "3213" "7662" "9313" "0000" "2222" "3333" "5555" "8193" "8096" "7777" "9999" "7756" "6855" "9881" "5531")))

(map classic input) ;-> (6 0 0 4 0 0 2 1 4 0 0 0 3 5 0 4 1 3 5 0)

;; What is:
(classic "2581")

;; What is classic ?

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Just to make it a bit easier to read

(map (juxt identity classic) input) ;-> 

["8809" 6]
["7111" 0]
["2172" 0]
["6666" 4]
["1111" 0]
["3213" 0]
["7662" 2]
["9313" 1]
["0000" 4]
["2222" 0]
["3333" 0]
["5555" 0]
["8193" 3]
["8096" 5]
["7777" 0]
["9999" 4]
["7756" 1]
["6855" 3]
["9881" 5]
["5531" 0]



Thursday, June 26, 2014

I Wish I'd Written This

Nothing specifically  to do with clojure, but I figure that a fair number of readers of this blog must love algorithms and visualizations, and this:

http://bost.ocks.org/mike/algorithms/

is one of the best things I've ever read on those subjects. I really wish I'd written this! Or even was capable of writing this.

Tuesday, May 20, 2014

Planes In Space : Random Sampling


;; My recursion tells me that n planes can divide space into
;; 2, 4, 8, 15, 26, and finally 42 regions.

;; But I'd like some sort of empirical confirmation of this result.

;; Forms in 3-space can be represented as ordered sets of three numbers
(def a-form [1 2 3])

;; As can vectors
(def a-point [2 3 5])

;; And we evaluate the form on the vector
(defn contract [form vector]
  (reduce + (map * form vector)))
;; (or if you prefer, take the dot product, or contract the tensors)

(contract a-form a-point) ;-> 23

;; Any plane can be defined in terms of a 1-form and a number.  
(def a-plane {:form [1 2 3] :val 4})

;; Now if we have a plane or a vector, we can evaluate the form on the
;; vector, and compare the result with the value. This tells us which
;; side of the plane the vector is on.

(defn side [plane point]
  (- (contract (plane :form) point) (plane :val)))


(side a-plane [2 3  5]) ;-> 19  (this point is on the positive side)
(side a-plane [2 3 -5]) ;-> -11 (on the negative side)
(side a-plane [2 3 -4/3]) ;-> 0N (in the plane itself)


;; Ok, now we need a way of taking vectors and forms at random. 
;; The cauchy distribution is easy to sample from and has nice fat tails

(defn cauchy[]
  (Math/tan (* Math/PI (- (rand) 0.5))))

(repeatedly 20 cauchy) ;-> (-0.43989542100474244 -0.6517139433588255 1.58518947555566 0.001268073580101198 3.6164981498788262 0.44928717656825584 0.3365831420089349 0.4646894211443393 0.8802485518044282 1.8146747880005754 0.1608864471756546 -0.23538854021056904 8.836583912257565 3.8174659241864703 0.5387819323291936 -0.18830386363467239 -1.0430272980416788 0.3310448308016341 -0.10735190850334911 0.3426157380908667)

(defn make-point []
  (repeatedly 3 cauchy))

(defn make-plane []
  {:form (repeatedly 3 cauchy) :val (cauchy)})

(make-point) ;-> (33.032354006369815 -29.428219536044043 -37.796430533111334)
(make-plane) ;-> {:form (-45.36910184399889 -1.6741101969009575 9.952054197916382), :val 0.9505471630252558}

(def points (repeatedly #(make-point)))
(def planes (repeatedly #(make-plane)))

;; And we'll need a function to tell us the sign of a number
(defn sign[x]
  (if (< x 0) '- '+))

(map sign [ -1 -2 -3 0 -0.5 1.3]) ;-> (- - - + - +)

;; Now if we take a set of planes and a point, 
(defn sig [point planes]
  (for [p planes] (sign (side p point))))

;; We can check which side of each plane the point is on
(sig (first points) (take 3 planes)) ;-> (+ - +)

;; Every different region gives a different signature. 
;; The more planes, the more signatures.

(count (frequencies (take 10 (map #(sig % (take 1 planes)) points)))) ;-> 2
(count (frequencies (take 10 (map #(sig % (take 2 planes)) points)))) ;-> 4
(count (frequencies (take 10 (map #(sig % (take 3 planes)) points)))) ;-> 6
(count (frequencies (take 10 (map #(sig % (take 4 planes)) points)))) ;-> 7
(count (frequencies (take 10 (map #(sig % (take 5 planes)) points)))) ;-> 7
(count (frequencies (take 10 (map #(sig % (take 6 planes)) points)))) ;-> 7

;; But the more planes we have, the smaller the smallest regions are
;; and thus the chance of a point falling in every one goes down.


;; The more points we take, the more likely we are to get one in every region

(count (frequencies (take 100 (map #(sig % (take 1 planes)) points)))) ;-> 2
(count (frequencies (take 100 (map #(sig % (take 2 planes)) points)))) ;-> 4
(count (frequencies (take 100 (map #(sig % (take 3 planes)) points)))) ;-> 7
(count (frequencies (take 100 (map #(sig % (take 4 planes)) points)))) ;-> 11
(count (frequencies (take 100 (map #(sig % (take 5 planes)) points)))) ;-> 18
(count (frequencies (take 100 (map #(sig % (take 6 planes)) points)))) ;-> 21

(count (frequencies (take 1000 (map #(sig % (take 1 planes)) points)))) ;-> 2
(count (frequencies (take 1000 (map #(sig % (take 2 planes)) points)))) ;-> 4
(count (frequencies (take 1000 (map #(sig % (take 3 planes)) points)))) ;-> 8
(count (frequencies (take 1000 (map #(sig % (take 4 planes)) points)))) ;-> 15
(count (frequencies (take 1000 (map #(sig % (take 5 planes)) points)))) ;-> 26
(count (frequencies (take 1000 (map #(sig % (take 6 planes)) points)))) ;-> 38

(count (frequencies (take 10000 (map #(sig % (take 1 planes)) points)))) ; 2 
(count (frequencies (take 10000 (map #(sig % (take 2 planes)) points)))) ; 4 
(count (frequencies (take 10000 (map #(sig % (take 3 planes)) points)))) ; 8 
(count (frequencies (take 10000 (map #(sig % (take 4 planes)) points)))) ; 15 
(count (frequencies (take 10000 (map #(sig % (take 5 planes)) points)))) ; 26 
(count (frequencies (take 10000 (map #(sig % (take 6 planes)) points)))) ; 40 

(count (frequencies (take 100000 (map #(sig % (take 1 planes)) points)))) ; 2 
(count (frequencies (take 100000 (map #(sig % (take 2 planes)) points)))) ; 4 
(count (frequencies (take 100000 (map #(sig % (take 3 planes)) points)))) ; 8 
(count (frequencies (take 100000 (map #(sig % (take 4 planes)) points)))) ; 15
(count (frequencies (take 100000 (map #(sig % (take 5 planes)) points)))) ; 26
(count (frequencies (take 100000 (map #(sig % (take 6 planes)) points)))) ; 41

(count (frequencies (take 1000000 (map #(sig % (take 1 planes)) points)))) ; 2
(count (frequencies (take 1000000 (map #(sig % (take 2 planes)) points)))) ; 4
(count (frequencies (take 1000000 (map #(sig % (take 3 planes)) points)))) ; 8
(count (frequencies (take 1000000 (map #(sig % (take 4 planes)) points)))) ; 15
(count (frequencies (take 1000000 (map #(sig % (take 5 planes)) points)))) ; 26
(count (frequencies (take 1000000 (map #(sig % (take 6 planes)) points)))) ; 42 

;; I'm painfully conscious of having stopped the experiment at the
;; exact point where I got the answer I expected. But my poor little
;; computer is not going to be up to running this for 10000000 points.

;; But this can't just be coincidence, surely?








Fizz Buzz : An Interview Question


;; Fizz Buzz

;; My sources
;; http://blog.codinghorror.com/why-cant-programmers-program/
;; inform me that:

;; The majority of computer science graduates can't solve this problem:

;; Write a program that prints the numbers from 1 to 100.  But for
;; multiples of three print "Fizz" instead of the number and for the
;; multiples of five print "Buzz". For numbers which are multiples of
;; both three and five print "FizzBuzz".

;; And Brother Downing of this parish, who actually hires people to
;; program in Java and Clojure learns me that he does indeed use this
;; to screen job applicants, and that most of them can't do it.

;; It is hard to read a thing like that without thinking: 'hang on, is that harder than it looks?'

;; So I did it, just to check:

;; I decided to use pull it out your ass driven development, where
;; you just pull the answer out of your ass.

;; First bit, print out the numbers from 1 to 100
(range 100) ;-> (0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 ...)

;; Bugger
(range 1 101) ;-> (1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 ...)

;; paranoid now
(last (range 1 101)) ;-> 100

;; Although I guess really (print (range 1 101)) is what I want
;; here. In PIOYADD, we defer this important user interface question for later.

;; Next, print 'Fizz' instead of all the multiples of three
(map (= 0 #(% quot 3)) (range 1 101))
;; ClassCastException java.lang.Boolean cannot be cast to clojure.lang.IFn  clojure.core/map/fn--4207 (core.clj:2485)

;; bugger
(map #(= 0 (quot % 3)) (range 1 101)) ;-> (true true false false false false false false false false false false false false false false false false false false false false false false false false false ...)

;; ok
(map #(if (= 0 (quot % 3)) "Fizz" %) (range 1 101)) ;-> ("Fizz" "Fizz" 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 ...)

;; oh for fuck's sake
(map #(if (= 0 (mod % 3)) "Fizz" %) (range 1 101)) ;-> (1 2 "Fizz" 4 5 "Fizz" 7 8 "Fizz" 10 11 "Fizz" 13 14 "Fizz" 16 17 "Fizz" 19 20 "Fizz" 22 23 "Fizz" 25 26 "Fizz" ...)

;; payday
(map #(case (= 0 (mod % 3)) "Fizz" (= 0 (mod % 5)) "Buzz" %) (range 1 101))
;; IllegalArgumentException No matching clause: false  user/eval1234/fn--1235 (NO_SOURCE_FILE:1)

;; ok, I always screw that up
(map #(cond (= 0 (mod % 3)) "Fizz" (= 0 (mod % 5)) "Buzz" %) (range 1 101))
;; CompilerException java.lang.IllegalArgumentException: cond requires an even number of forms, compiling:(NO_SOURCE_PATH:1:7) 

;; twice usually
(map #(cond (= 0 (mod % 3)) "Fizz" (= 0 (mod % 5)) "Buzz" :else %) (range 1 101)) 
;-> (1 2 "Fizz" 4 "Buzz" "Fizz" 7 8 "Fizz" "Buzz" 11 "Fizz" 13 14 "Fizz" 16 17 "Fizz" 19 "Buzz" "Fizz" 22 23 "Fizz" "Buzz" 26 "Fizz" ...)

;; Bwahhahhhahhh! No, Mr Bond, I expect you to die.

;; And now, for the tricky bit of the problem, we unsheathe the
;; superweapon, copy-and-paste-driven development
(map #(cond (or (= 0 (mod % 3)) (= 0 (mod % 3))) "FizzBuzz" (= 0 (mod % 5)) "Buzz" :else %) (range 1 101)) 
;-> (1 2 "FizzBuzz" 4 "Buzz" "FizzBuzz" 7 8 "FizzBuzz" "Buzz" 11 "FizzBuzz" 13 14 "FizzBuzz" 16 17 "FizzBuzz" 19 "Buzz" "FizzBuzz" 22 23 "FizzBuzz" "Buzz" 26 "FizzBuzz" ...)

;; hmmm
(map #(cond (and (= 0 (mod % 3)) (= 0 (mod % 3))) "FizzBuzz" (= 0 (mod % 5)) "Buzz" :else %) (range 1 101)) 
;-> (1 2 "FizzBuzz" 4 "Buzz" "FizzBuzz" 7 8 "FizzBuzz" "Buzz" 11 "FizzBuzz" 13 14 "FizzBuzz" 16 17 "FizzBuzz" 19 "Buzz" "FizzBuzz" 22 23 "FizzBuzz" "Buzz" 26 "FizzBuzz" ...)

;; still wrong
(map #(cond (and (= 0 (mod % 3)) (= 0 (mod % 5))) "FizzBuzz" (= 0 (mod % 5)) "Buzz" :else %) (range 1 101)) 
;-> (1 2 3 4 "Buzz" 6 7 8 9 "Buzz" 11 12 13 14 "FizzBuzz" 16 17 18 19 "Buzz" 21 22 23 24 "Buzz" 26 27 ...)

;; aargh, where did the fizzes go?
(map #(cond (and (= 0 (mod % 3)) (= 0 (mod % 5))) "FizzBuzz" (= 0 (mod % 3)) "Fizz" (= 0 (mod % 5)) "Buzz" :else %) (range 1 101)) 
;-> (1 2 "Fizz" 4 "Buzz" "Fizz" 7 8 "Fizz" "Buzz" 11 "Fizz" 13 14 "FizzBuzz" 16 17 "Fizz" 19 "Buzz" "Fizz" 22 23 "Fizz" "Buzz" 26 "Fizz" ...)

;; Wahey! That looks done. Three minutes.

;; And now, pretending you're not a filthy hacker driven development:
(map 
 #(cond (and (= 0 (mod % 3)) (= 0 (mod % 5))) "FizzBuzz" 
        (= 0 (mod % 3)) "Fizz" 
        (= 0 (mod % 5)) "Buzz" 
        :else %) (range 1 101)) 
;-> (1 2 "Fizz" 4 "Buzz" "Fizz" 7 8 "Fizz" "Buzz" 11 "Fizz" 13 14 "FizzBuzz" 16 17 "Fizz" 19 "Buzz" "Fizz" 22 23 "Fizz" "Buzz" 26 "Fizz" ...)

(defn divides? [n m] (= 0 (mod n m)))
(divides? 3 15) ;-> false

;; sigh
(defn divides? [n m] (= 0 (mod m n)))
(divides? 3 15) ;-> true
(divides? 15 3) ;-> false
;; rah!

;; and so, behold: beauty is truth, and truth, beauty
(map 
 #(cond (divides? 15 %) "FizzBuzz" 
        (divides? 3 %)  "Fizz" 
        (divides? 5 %)  "Buzz" 
        :else %) (range 1 101)) 
;-> (1 2 "Fizz" 4 "Buzz" "Fizz" 7 8 "Fizz" "Buzz" 11 "Fizz" 13 14 "FizzBuzz" 16 17 "Fizz" 19 "Buzz" "Fizz" 22 23 "Fizz" "Buzz" 26 "Fizz" ...)

;; finally, smugness driven development:
(def fizzbuzz
  (map 
   #(cond (divides? 15 %) "FizzBuzz" 
          (divides? 3 %)  "Fizz" 
          (divides? 5 %)  "Buzz" 
          :else %) (map inc (range))))

;; There are those who would call this 'premature abstraction', but they deserve not the names of men.
(print (take 100 fizzbuzz))
;; (1 2 Fizz 4 Buzz Fizz 7 8 Fizz Buzz 11 Fizz 13 14 FizzBuzz 16 17 Fizz 19 Buzz Fizz 22 23 Fizz Buzz 26 Fizz ...)

;; And I suppose a regression test would be nice, if I'm trying to
;; give some sort of professional impression:
(= 
 (take 27 fizzbuzz)
 (list 1 2 "Fizz" 4 "Buzz" "Fizz" 7 8 "Fizz" "Buzz" 
       11 "Fizz" 13 14 "FizzBuzz" 16 17 "Fizz" 19 "Buzz"
       "Fizz" 22 23 "Fizz" "Buzz" 26 "Fizz"))

;; And some paranoid checks, if I'm going to put this travesty on my blog:
(count (filter #(= "Fizz" %) (take 1000 fizzbuzz))) ; -> 267
(count (filter #(= "FizzBuzz" %) (take 1000 fizzbuzz))) ;-> 66
(count (filter #(= "Buzz" %) (take 1000 fizzbuzz))) ;-> 134

(* (+ 134 66) 5) ;-> 1000
(* (+ 267 66) 3) ;-> 999

;; bah, that's close enough for government work. I declare myself
;; done. Three minutes of flail and two minutes of tidying up and
;; checking it works.

;; So my question to the wider community is: Does my three minutes of
;; flailing trying to remember the semantics of my favourite language
;; (which I can perfectly imagine looking dreadful at an interview)
;; count as a fail or a pass?

;; In a previously unknown assembly language, but given a library to
;; print numbers on the screen, I can imagine taking half an hour to
;; get this working. Without the library, it's a research project.

;; And obviously, if I tried to do it in Haskell, I'd have to spend
;; three weeks remembering what a monad was in order to print the
;; damned thing out, but at least the type system would magically
;; guarantee the correctness of the final program.

;; In other words, is fizzbuzz really actually quite hard, or is
;; everyone out there a complete idiot?






Monday, May 12, 2014

Planes in Space : Checking by Hand


;; Ok, so I reckon:

(defn regions [n m]
  (cond (= n 1) (list m (inc m))
        (= m 0) (concat (repeat n 0) '(1))
        :else (map + 
                   (regions n (dec m))
                   (concat '(0) (regions (dec n) (dec m))) 
                   (concat (regions (dec n) (dec m)) '(0)))))

;; In fact, I reckon that I can prove it. 

;; It's quite easy to prove something that isn't true. Usually when
;; you run across a counterexample, that shows you where the
;; unsuspected false step in your reasoning was.

;; So I'd like to verify the formula on lots of examples.

;; But how?

;; These I can count in my head.

(regions 3 0) ;-> (0 0 0 1) ; just space, no planes
(regions 3 1) ;-> (0 0 1 2) ; a plane cuts space in half
(regions 3 2) ;-> (0 1 4 4) ; two planes intersect in one line, cutting space into four
(regions 3 3) ;-> (1 6 12 8) ; 1 point where three planes meet, 
                             ; six coordinate half-axes, 
                             ; three coordinate planes divided into four quadrants each,
                             ; and eight octants

;; I am sort of confident that four planes make fifteen regions and intersect in four points.
(regions 3 4) ;-> (4 18 28 15)
;; But every attempt to count the 18 lines and 28 regions I make ends up relying on 
;; the sort of arguments I made to make the recursion in the first place.

;; And at this point my intuition breaks.
(regions 3 5) ;-> (10 40 55 26)

;; I mean, five planes, any three intersect in a point,
;; 10 ways to choose 3 planes from five, so 10 points,
;; but after that I'm dead.

;; And this? Six choose 3 is 20, I can see that....
(regions 3 6) ;-> (20 75 96 42)

;; And I defy anyone to even picture
(regions 5 8) ;-> (56 350 896 1176 792 219)

;; 8 choose 5 is (/ (* 8 7 6) (* 1 2 3)) = 56, which is fair enough.

;; 8 choose 4 is (/ (* 8 7 6 5) (* 1 2 3 4)) is 70

;; If we take any 4 hyperplanes from our 8, they'll define a line.
;; The four remaining hyperplanes then divide each line like:
(regions 1 4) ;-> (4 5)
;; So if each of those lines is sliced into 5 pieces then that's our 350.
;; But I'm just using the same recursive argument again.

;; So I don't even know if that's evidence or not.

;; One nice thing about it, the formula has an alternating sum property, 
;; like the Euler index.

(defn signature [lst]
  (reduce + 
          (map * 
               (apply concat (repeat '(+1 -1))) 
               (reverse lst))))

(signature (regions 5 8)) ;-> 1

(def regions (memoize regions))
(regions 7 23) ;-> (245157 1817046 5787628 10271800 10973116 7057688 2531288 390656)
(signature (regions 7 23)) ;-> 1

(regions 23 20) ;-> (0 0 0 1 40 760 9120 77520 496128 2480640 9922560 32248320 85995520 189190144 343982080 515973120 635043840 635043840 508035072 317521920 149422080 49807360 10485760 1048576)
(signature (regions 23 20)) ;-> 1

;; But annoyingly, you can just read that property straight from the recursion!

Wednesday, April 30, 2014

Planes in Spaaace!


;; Into how many regions can six planes divide space?

;; Too hard!

;; So make it harder:

;; Into how many regions can m hyperplanes divide n dimensional space?

;; Now look at the easiest case.

;; Into how many lines can m points divide a line?

;; We start off with one line and no points

;; n = 1 
;; m  points lines
;; 0  0      1

;; Every time we add a point, it splits a line

;; n = 1 
;; m  points lines
;; 0  0      1
;; 1  1      2
;; 1  1      2

(defn regions [n m]
  (cond (= n 1) (list m (inc m))
        :else 'chsaispas))

(regions 1 10) ;-> (10 11)
;; Adding ten points turns a line into 11 lines and 10 points.

;; Ok, what about lines dividing a plane?

;; We start off with one area, no lines and no points.

;; n = 2 
;; m  points lines areas
;; 0  0      0     1

;; When we add a line, it splits the area into two areas, so we have
;; no points, one line and two areas

;; n = 2 
;; m  points lines areas
;; 0  0      0     1
;; 1  0      1     2

;; When we add a second line, unless we've accidentally put it
;; parallel to our first line, in which case we aren't doing as well
;; as we could do, it will cross the first line.

;; So our new line will be actually be one point and two lines, 
;; and the one point will divide the first line in two, 
;; and the two lines will cut the two areas into four.

;; 0      1     2  ;; State with one line
;; 1      2        ;; Added one point and two lines
;;        1     2  ;; Which have split one line and two areas
;; 1      4     4  ;; So two lines divide a plane into one point, four lines and four areas

;; n = 2 
;; m  points lines areas
;; 0  0      0     1
;; 1  0      1     2
;; 2  1      4     4

;; When we add a third line, it will cross both previous lines, and so it will be 
;; a line divided by two points 

(regions 1 2) ;-> (2 3)

(map + '(1 4 4) 
     (concat '(0) (regions 1 2)) 
     (concat (regions 1 2) '(0))) ;-> (3 9 7)

;; If you draw the figure, you'll see that it indeed has a triangle with three vertices, 
;; which split the three lines into 9, and there are seven distinct regions.

;; So, taking a blind leap in the dark...

(defn regions [n m]
  (cond (= n 1) (list m (inc m))
        (= m 0) (concat (repeat n 0) '(1))
        :else (map + 
                   (regions n (dec m))
                   (concat '(0) (regions (dec n) (dec m))) 
                   (concat (regions (dec n) (dec m)) '(0)))))

(regions 1 0) ;-> (0 1)
(regions 1 1) ;-> (1 2)
(regions 1 2) ;-> (2 3)
(regions 1 3) ;-> (3 4)

(regions 2 0) ;-> (0 0 1)
(regions 2 1) ;-> (0 1 2)
(regions 2 2) ;-> (1 4 4)
(regions 2 3) ;-> (3 9 7)
(regions 2 4) ;-> (6 16 11)
(regions 2 5) ;-> (10 25 16)

(regions 3 1) ;-> (0 0 1 2)
(regions 3 2) ;-> (0 1 4 4)
(regions 3 3) ;-> (1 6 12 8)
(regions 3 4) ;-> (4 18 28 15)
(regions 3 5) ;-> (10 40 55 26)
(regions 3 6) ;-> (20 75 96 42)

;; Into how many regions can six planes divide space?

(last (regions 3 6)) ;-> 42



Into How Many Regions Can Six Planes Divide Space?

I recently misread an easy recreational puzzle as 'Into how many regions can six planes divide space?'.

It took me about half an hour to get an answer. (Which was the correct answer to the question, but not the answer to the real puzzle). But it turned out to be great fun to think about.

I'm pretty sure that my answer's correct (and wikipedia agrees), but although the combination of mathematical proof and worldwide consensus should probably be enough, I'm never really convinced by a mathematical idea until I've seen it confirmed by lots of examples.

So I've got programs planned, and if they turn out well, then I'll post them here. But they won't be any fun to read unless you've tried to work it out yourself.

If it's not clear, the idea is that you've got a big block of sponge cake, and a very large knife, and you want to cut it into as many pieces as possible with six cuts.

So if the problem said 'one' rather than 'six', the answer would be two, and if it said 'two', then the answer would be 'four'.

1,2,4,8,..... how does this sequence continue?

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