Generating Palindromes

August 29, 2014

The first hundred palindromic numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 606, 616, 626, 636, 646, 656, 666, 676, 686, 696, 707, 717, 727, 737, 747, 757, 767, 777, 787, 797, 808, 818, 828, 838, 848, 858, 868, 878, 888, 898, and 909.

Your task is to write a program that generates the palindromic numbers in order; use it to find the ten-thousandth palindromic number. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Integer Logarithm

August 26, 2014

The integer logarithm function in the Standard Prelude looks like this:

(define (ilog b n)
  (let loop1 ((lo 0) (b^lo 1) (hi 1) (b^hi b))
    (if (< b^hi n) (loop1 hi b^hi (* hi 2) (* b^hi b^hi))
      (let loop2 ((lo lo) (b^lo b^lo) (hi hi) (b^hi b^hi))
        (if (<= (- hi lo) 1) (if (= b^hi n) hi lo)
          (let* ((mid (quotient (+ lo hi) 2))
                 (b^mid (* b^lo (expt b (- mid lo)))))
            (cond ((< n b^mid) (loop2 lo b^lo mid b^mid))
                  ((< b^mid n) (loop2 mid b^mid hi b^hi))
                  (else mid))))))))

It performs binary search in loop1 until n is bracketed between b^lo and b^hi, doubling at each step, then refines the binayr search in loop2, halving the bracket at each step.

Inspired by that function, Joe Marshall posed this puzzle at his Abstract Heresies web site:

You can get the most significant digit (the leftmost) of a number pretty quickly this way:

(define (leftmost-digit base n)
  (if (< n base)
      n
      (let ((leftmost-pair (leftmost-digit (* base base) n)))
        (if (< leftmost-pair base)
            leftmost-pair
            (quotient leftmost-pair base)))))

The puzzle is to adapt this code to return the position of the leftmost digit.

Your task is to write Joe’s puzzle function; you might also click through to his website to spike his stats. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Patience Sorting

August 22, 2014

This sorting algorithm derives its name from the game of Patience (that’s the British name, we call it Solitaire in the United States) because it is implemented by analogy to sorting a shuffled deck of cards:

Starting with no piles, add the next card from the deck to the first pile with top card greater than the next card from the deck. If the next card from the deck is greater than the top card of all the piles, start a new pile. When the deck is exhausted, collect the cards in order by selecting the smallest visible card at each step.

For instance, consider sorting the list (4 3 9 1 5 2 7 8 6). The first stack gets 4 and 3. Since 9 is larger than 3, it starts a second stack, 1 goes on the first stack, then 5 and 2 go on the second stack. At this point the first stack (top to bottom) consists of (1 3 4), the second stack consists of (2 5 9), and the remaining deck consists of (7 8 6). Now 7 goes on a third stack, 8 goes on a fourth stack, and 6 goes on top of the 7 in the third stack. With all the cards dealt, 1 is collected from the first stack, 2 from the second stack, 3 and 4 from the first stack, 5 from the second stack, 6 and 7 from the third stack, 8 from the fourth stack, and 9 from the second stack. The algorithm has complexity O(n log n).

Your task is to implement the patience sorting algorithm. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Cracker Barrel

August 19, 2014

While I was out of town last week, I ate dinner one evening at Cracker Barrel, which provides a triangle puzzle at each table so you can amuse yourself while you are waiting for your food. When my daughter challenged me to solve the problem, I failed, but I promised her that I would write a program to solve it.

As shown in the picture at right, the puzzle is a triangle with 15 holes and 14 pegs; one hole is initially vacant. The game is played by making 13 jumps; each jump removes one peg from the triangle, so at the end of 13 jumps there is one peg remaining. A jump takes a peg from a starting hole, over an occupied hole, to an empty finishing hole, removing the intermediate peg.

Your task is to write a program that solves the Cracker Barrel puzzle; find all possible solutions that start with a corner hole vacant and end with the remaining peg in the original vacant corner. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Today’s exercise is a frequent source of questions at places like Stack Overflow and /r/learnprogramming; it must come from one of the competitive programming sites like SPOJ or UVA. The most common statement of the problem is something like this:

You are given two positive integers P and Q, either of which can be quite large, up to a million digits. Compute PQ, that is, P raised to the Q power. For your convenience, give the result modulo 109 + 7. For instance, with P = 34534985349875439875439875349875 and Q = 93475349759384754395743975349573495, the expected result is 735851262.

The phrase “for your convenience” is a giveaway that the modulo computation is a trick; that kind of contest site never does anything for your convenience. In fact, due to a corollary of Fermat’s little theorem, PQ (mod m) ≡ pq (mod m) where p = P (mod m) and q = Q (mod m − 1). That makes it easy. Compute p and q. Both will be less than 232, so we can use the Montgomery multiplication algorithm of a previous exercise to make the computation.

Your task is to write a program to perform the big modular exponentiations described above. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Minimal Palindromic Base

August 5, 2014

We have a simple little problem today: Given an integer n > 2, find the minimum b > 1 for which n base b is a palindrome. For instance, n = 15 = 11112 = 1203 = 334 = 305 = 236 = 217 = 178 = 169 = 1510 = 1411 = 1312 = 1213 = 1114; of those, bases 2, 4 and 14 form palindromes, and the least of those is 2, so the correct answer is 2.

Your task is to write a program that calculates the smallest base for which a number n is a palindrome. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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K’th Largest Item

August 1, 2014

Today’s exercise is a common interview question, and also interesting in its own right:

Find the kth largest integer in a file of n integers, where n is large (defined as too large to fit into memory all at once) and k is small (defined as small enough to all fit in memory).

Your task is to write the program described above. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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