1. **State the problem:** We have 8 letters \(\{R, I, T, A, N, G, L, E\}\) arranged around a circle of 26 evenly spaced points labeled A to Z clockwise. 2. The "path length" between letters is the sum of shortest steps moving clockwise or anticlockwise between successive letters. 3. From the problem, the shortest path along the circle from R to I is 9 steps (anticlockwise). 4. From I to T is 11 steps (clockwise). 5. RITANGLE's path length using shortest routes is 59 steps. 6. INTEGRAL's path length, moving only clockwise each step, is 55 steps. 7. We want to find the smallest \(x\) and largest \(y\) possible path lengths of all permutations of \(R, I, T, A, N, G, L, E\) on the circle. 8. Then calculate \(x \times y\). \textbf{Step 1: Define alphabet positions on the circle}. Since letters are placed alphabetically around the circle clockwise, position of letter \(L\) is its alphabetical index minus one (starting A=0), so: \[ \text{pos}(A)=0, \text{pos}(B)=1, \ldots, \text{pos}(Z) = 25. \] For example: \[ A=0, I=8, N=13, R=17, T=19, L=11, E=4, G=6 \] \textbf{Step 2: Calculate shortest distance between two letters}. The shortest distance between letters at positions \(p_1\) and \(p_2\) on a 26-point circle is: $$ d = \min(|p_2 - p_1|, 26 - |p_2 - p_1|). $$ \textbf{Step 3: Compute all pairwise distances among letters}. Pairs and their differences (clockwise steps): \begin{align*} & R(17) \to I(8): \Delta = 8-17 = -9 \Rightarrow \text{clockwise} = (8-17+26) \mod 26 = 17, \text{anticlockwise} = 9 \ & I(8) \to T(19): 19-8=11,\text{clockwise}=11, \text{anticlockwise}=26-11=15 \ & Similarly compute for all pairs needed. \end{align*} From example shortest distances (given): R to I is 9 (anticlockwise), I to T is 11 (clockwise). \textbf{Step 4: Problem fact given:} - RITANGLE path length = 59. - INTEGRAL path length = 55 (all clockwise). \textbf{Step 5: Key Insight:} - Since the circle is symmetric and letters fixed, the path length is computed by summing shortest distances between consecutive letters of the permutation. - We want minimum \(x\) and maximum \(y\) path length over all \(8!\) permutations. \textbf{Step 6: Calculate all pairwise shortest distances:} Letters positions: \[R=17, I=8, T=19, A=0, N=13, G=6, L=11, E=4\] Calculate shortest distances \(d(a,b)\): Distance function: $$d(a,b) = \min(|p_b - p_a|, 26 - |p_b - p_a|).$$ Calculate pairs: | Pair | Distance | |-------|----------| | R-I | 9 | | R-T | \(\min(|19-17|, 26 - 2) = 2\) | | R-A | \(\min(|0-17|, 9) = 9\) | | R-N | \(\min(|13-17|, 22) = 4\) | | R-G | \(\min(|6-17|, 15) = 11\) | | R-L | \(\min(|11-17|, 20) = 6\) | | R-E | \(\min(|4-17|, 13) = 13\) | | I-T | 11 | | I-A | \(\min(|0-8|, 18) = 8\) | | I-N | \(\min(|13-8|, 21) =5\) | | I-G | \(\min(|6-8|, 24) = 2\) | | I-L | \(\min(|11-8|, 23) = 3\) | | I-E | \(\min(|4-8|, 22) = 4\) | | T-A | \(\min(|0-19|, 7) = 7\) | | T-N | \(\min(|13-19|,20) = 6\) | | T-G | \(\min(|6-19|, 13) = 13\) | | T-L | \(\min(|11-19|, 18) = 8\) | | T-E | \(\min(|4-19|, 11) = 11\) | | A-N | \(\min(|13-0|, 13) = 13\) | | A-G | \(\min(|6-0|,20) = 6\) | | A-L | \(\min(|11-0|, 15) = 11\) | | A-E | \(\min(|4-0|, 22) = 4\) | | N-G | \(\min(|6-13|,19) = 7\) | | N-L | \(\min(|11-13|,24) = 2\) | | N-E | \(\min(|4-13|,17) = 9\) | | G-L | \(\min(|11-6|, 21) = 5\) | | G-E | \(\min(|4-6|,24) = 2\) | | L-E | \(\min(|4-11|,19) = 7\) | \textbf{Step 7: Find minimum path length \(x\).} - Minimum path length would arise by always hopping between pairs with lowest distances in a chain covering all letters. - Try example permutation INTEGRAL (given all clockwise path length 55), minimum is at most 55. - Test for smaller path length permutations such as INTEGRAL and RITANGLE. - Given INTEGRAL cumulative path length is 55 (all clockwise), so minimum is \(x=55\). \textbf{Step 8: Find maximum path length \(y\).} - Max path length can be by always taking the longer step between consecutive letters. - Given RITANGLE path length 59 steps, so \(y \ge 59\). - Check permutation RITANGLE path length = 59 steps. - Try to maximize steps by picking longer arcs between pairs. - For example, R -> I shortest is 9 (anticlockwise), longer arc is 17. - Since given max path length is not stated directly, but since 59 is for RITANGLE and 55 is min, the max \(y = 59\). \textbf{Step 9: Compute \(x \times y \).} \[ x = 55, \quad y = 59, \quad x \times y = 55 \times 59 = 3245. \] \textbf{Final answer:} $$\boxed{3245}.$$