Problem 4: Largest Palindrome Product

A palindromic number reads the same forward and backward. For two-digit factors the maximum example is $9009 = 91 im es 99$ , showcasing how symmetry appears in multiplication.

For a general digit length $n$ , we examine factor pairs between $hi = 1 0^{n} - 1$ and $l o = 1 0^{n - 1}$ to locate the greatest palindromic product.

How to Solve

Descend $i$ from $hi = 1 0^{n} - 1$ to $l o = 1 0^{n - 1}$ . For each outer value, walk $j$ downward and exit early whenever $i im es hi \leq e x t b es t$ because no larger palindrome can appear for this $i$ .

Every even-length palindrome is divisible by $11$ , so if $i$ is not a multiple we only test $j$ values that are. This reduces the inner loop dramatically without missing any candidates.

Iterate $j$ down to $i$ to avoid duplicate pairs. The first time $i im es j$ reads the same forwards and backwards, update the best answer, capture the factors, and break the inner loop before continuing.

Code Solution


function largestPalindromeProduct(n) {
  const hi = Math.pow(10, n) - 1;
  const lo = Math.pow(10, n - 1);
  let best = 0;

  for (let i = hi; i >= lo; i--) {
    if (i * hi <= best) break;

    let jStart, jStep;
    if (i % 11 === 0) {
      jStart = hi;
      jStep = 1;
    } else {
      jStart = hi - (hi % 11);
      jStep = 11;
    }

    for (let j = jStart; j >= i; j -= jStep) {
      const prod = i * j;
      if (prod <= best) break;

      const s = String(prod);
      if (s === s.split("").reverse().join("")) {
        best = prod;
        break;
      }
    }
  }
  return best;
}

Test the Solution

Digits (n):

Interaction Example (static) — n = 3

Step	i	jStart	jStep	j	prod = i·j	isPalindrome	best	Comments
1	999	990	11	990	989010	❌	0	i%11≠0 → test j multiples of 11
2	999	990	11	979	978021	❌	0	not palindrome
3	999	990	11	968	967032	❌	0	continue
4	998	990	11	990	988020	❌	0	new i; still multiples of 11 for j
5	997	990	11	990	987030	❌	0	continue
6	996	990	11	990	986040	❌	0	continue
7	995	990	11	990	985050	❌	0	continue
8	994	990	11	990	984060	❌	0	continue
9	993	990	11	990	983070	❌	0	start i=993
10	993	990	11	979	972147	❌	0	not palindrome
11	993	990	11	968	961224	❌	0	not palindrome
12	993	990	11	957	950301	❌	0	not palindrome
13	993	990	11	946	939378	❌	0	not palindrome
14	993	990	11	935	928455	❌	0	not palindrome
15	993	990	11	924	917532	❌	0	not palindrome
16	993	990	11	913	906609	✅	906609	palindrome found → best = 906609 (993×913)

✅ For n = 3 the first palindrome hit shown here is 906609 from i = 993 and j = 913.

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