Can You Shape the Peloton?

Fiddler

Earlier this month, I watched La Vuelta, one of the three grand tours of cycling (a trio that includes the Tour de France). I noticed the peloton—the main group of riders—took on an aerodynamic profile that sometimes looked like a triangle, sometimes looked like a rhombus, and sometimes looked somewhere in between.

For example, the figure below shows the four possible formations between a triangle and a rhombus when the peloton’s maximum width is four riders:

For certain numbers of riders, multiple formations like these are possible. In particular, there are two formations with 15 riders: a triangle that’s five riders wide at the base and an almost-rhombus that’s four riders wide in the middle, but missing the bottommost rider, as shown below.

After 15, what is the next smallest number of riders that similarly has two distinct formations between a triangle and a rhombus?

(Note: As I’m writing this, the sequence of numbers for which there are at least two arrangements does not appear to be on OEIS. I smell an opportunity here…)

Solution

Rhombi are perfect squares.

Note that all formations are some triangular number (including $0$) subtracted from a rhombus.

Note that the $k^{th}$ triangular number is the $(k-1)^{st}$ triangular number subtracted from the $k^{th}$ rhombus.

My code started with a list of the first $n$ rhombi and $n-1$ triangles:

  1. For each rhombus from the $1^{st}$ to $n^{th}$ :
    • Subtract the $0^{th}$ triangular number form it, and it to the list of pelotons.
    • Increase triangular number by one and repeat, the last being the $(n-1)^{st}$ triangular numbers.
  2. Create a frequency table from all pelotons in the list, sorted by frequency then value.
  3. A value, $v$, can be easily extracted for a frequency.
  4. The largest possible rhombus, $r$ is slightly less than twice a triangle of that value.
  5. For each rhombus from the $1^{st}$ to $r^{th}$:
    • Subtract the $0^{th}$ triangular number form it, identify if it is equal to the peloton value $v$.
    • Increase triangular number by one and repeat, the last being the $(r-1)^{st}$ triangular numbers.

Answer

36 can be expressed as two formations:

Here is a list of the first 1000 numbers with exactly two distinct pelotons:

15, 36, 43, 49, 64, 66, 78, 85, 99, 100, 118, 120, 134, 151, 159, 168, 169, 204, 210, 211, 219, 225, 241, 246, 256, 270, 274, 279, 283, 295, 309, 321, 323, 325, 345, 351, 355, 358, 364, 372, 376, 379, 386, 393, 394, 400, 405, 406, 423, 429, 438, 440, 456, 474, 498, 505, 508, 519, 523, 526, 528, 531, 540, 555, 559, 561, 570, 575, 576, 589, 593, 648, 651, 661, 664, 666, 670, 673, 693, 701, 708, 721, 739, 748, 750, 763, 764, 771, 774, 780, 790, 805, 808, 813, 814, 820, 834, 855, 856, 861, 879, 883, 890, 897, 899, 900, 903, 904, 916, 918, 933, 934, 940, 958, 960, 969, 979, 984, 988, 996, 1003, 1009, 1015, 1018, 1020, 1035, 1044, 1051, 1061, 1065, 1069, 1072, 1079, 1089, 1093, 1116, 1128, 1134, 1135, 1143, 1146, 1153, 1156, 1159, 1176, 1191, 1205, 1213, 1216, 1222, 1225, 1233, 1245, 1249, 1254, 1273, 1290, 1291, 1296, 1303, 1306, 1314, 1341, 1348, 1353, 1358, 1359, 1368, 1369, 1375, 1378, 1389, 1401, 1408, 1413, 1416, 1430, 1450, 1455, 1485, 1498, 1506, 1510, 1511, 1520, 1521, 1524, 1528, 1534, 1540, 1545, 1554, 1555, 1576, 1579, 1585, 1594, 1596, 1597, 1636, 1639, 1647, 1651, 1653, 1659, 1674, 1678, 1680, 1705, 1713, 1728, 1740, 1743, 1744, 1749, 1764, 1765, 1771, 1774, 1783, 1794, 1800, 1816, 1821, 1828, 1831, 1834, 1835, 1839, 1840, 1843, 1849, 1858, 1863, 1870, 1881, 1884, 1885, 1889, 1891, 1898, 1906, 1935, 1936, 1947, 1953, 1963, 1970, 1980, 1989, 1996, 2010, 2019, 2024, 2025, 2035, 2055, 2068, 2074, 2080, 2089, 2094, 2106, 2115, 2122, 2125, 2133, 2136, 2139, 2164, 2168, 2175, 2181, 2188, 2191, 2205, 2206, 2208, 2209, 2211, 2213, 2214, 2236, 2250, 2259, 2269, 2281, 2283, 2290, 2298, 2301, 2303, 2304, 2310, 2316, 2325, 2326, 2329, 2344, 2353, 2356, 2359, 2373, 2374, 2380, 2388, 2391, 2400, 2403, 2409, 2413, 2429, 2430, 2446, 2451, 2455, 2464, 2465, 2472, 2473, 2479, 2481, 2484, 2485, 2494, 2496, 2497, 2500, 2510, 2514, 2533, 2535, 2538, 2544, 2546, 2578, 2584, 2586, 2598, 2599, 2608, 2616, 2626, 2638, 2654, 2659, 2661, 2668, 2671, 2685, 2694, 2698, 2703, 2704, 2718, 2731, 2740, 2745, 2769, 2773, 2775, 2778, 2780, 2803, 2808, 2809, 2811, 2815, 2820, 2838, 2850, 2860, 2888, 2898, 2899, 2901, 2905, 2906, 2913, 2924, 2926, 2941, 2947, 2949, 2970, 2973, 2980, 2991, 2997, 3000, 3004, 3015, 3016, 3018, 3024, 3025, 3058, 3061, 3064, 3070, 3075, 3088, 3091, 3103, 3104, 3111, 3126, 3133, 3135, 3144, 3150, 3154, 3158, 3160, 3165, 3189, 3194, 3211, 3214, 3221, 3234, 3235, 3246, 3271, 3275, 3276, 3279, 3286, 3291, 3298, 3309, 3313, 3315, 3316, 3319, 3321, 3328, 3330, 3343, 3347, 3349, 3354, 3364, 3376, 3383, 3390, 3396, 3403, 3408, 3410, 3414, 3415, 3430, 3438, 3453, 3460, 3473, 3484, 3486, 3495, 3499, 3501, 3531, 3534, 3536, 3543, 3545, 3550, 3559, 3564, 3591, 3599, 3600, 3613, 3618, 3625, 3630, 3631, 3634, 3643, 3644, 3653, 3655, 3676, 3685, 3690, 3693, 3699, 3706, 3711, 3715, 3724, 3726, 3729, 3738, 3739, 3753, 3760, 3766, 3771, 3772, 3786, 3789, 3795, 3798, 3806, 3808, 3819, 3826, 3828, 3843, 3849, 3858, 3860, 3874, 3886, 3891, 3900, 3910, 3913, 3921, 3925, 3933, 3941, 3949, 3954, 3959, 3960, 3972, 3976, 3985, 3991, 3993, 3994, 4003, 4006, 4029, 4030, 4039, 4051, 4056, 4065, 4068, 4075, 4080, 4083, 4089, 4093, 4103, 4120, 4131, 4138, 4146, 4194, 4197, 4201, 4215, 4218, 4222, 4225, 4248, 4251, 4258, 4261, 4265, 4276, 4278, 4281, 4300, 4320, 4333, 4339, 4341, 4348, 4353, 4369, 4371, 4375, 4383, 4384, 4393, 4398, 4423, 4434, 4435, 4444, 4446, 4450, 4461, 4465, 4468, 4471, 4474, 4481, 4486, 4504, 4509, 4513, 4518, 4544, 4545, 4546, 4549, 4554, 4558, 4571, 4579, 4590, 4603, 4608, 4614, 4621, 4623, 4624, 4626, 4635, 4641, 4644, 4654, 4669, 4670, 4690, 4706, 4710, 4716, 4719, 4733, 4740, 4751, 4753, 4758, 4761, 4764, 4768, 4778, 4795, 4801, 4831, 4840, 4851, 4859, 4860, 4864, 4873, 4879, 4884, 4911, 4936, 4948, 4950, 4956, 4969, 4978, 4994, 4995, 4996, 5020, 5026, 5030, 5031, 5035, 5038, 5040, 5048, 5050, 5064, 5071, 5076, 5079, 5097, 5098, 5109, 5118, 5119, 5121, 5125, 5129, 5145, 5146, 5149, 5151, 5160, 5163, 5174, 5176, 5178, 5183, 5184, 5188, 5200, 5209, 5226, 5247, 5263, 5266, 5272, 5274, 5284, 5300, 5304, 5314, 5319, 5323, 5325, 5326, 5328, 5329, 5334, 5336, 5349, 5350, 5356, 5365, 5370, 5380, 5385, 5404, 5410, 5418, 5421, 5431, 5440, 5448, 5458, 5460, 5466, 5472, 5475, 5476, 5500, 5505, 5506, 5538, 5539, 5545, 5553, 5559, 5563, 5566, 5571, 5575, 5596, 5599, 5604, 5614, 5619, 5622, 5625, 5643, 5653, 5659, 5664, 5671, 5676, 5689, 5698, 5706, 5710, 5713, 5730, 5734, 5739, 5740, 5748, 5754, 5755, 5758, 5761, 5766, 5770, 5778, 5781, 5796, 5799, 5808, 5831, 5839, 5853, 5856, 5858, 5866, 5880, 5884, 5895, 5899, 5904, 5905, 5908, 5914, 5916, 5928, 5931, 5935, 5944, 5948, 5973, 5975, 5979, 5983, 5986, 5993, 5995, 6000, 6006, 6018, 6022, 6029, 6056, 6063, 6069, 6075, 6088, 6091, 6093, 6099, 6105, 6118, 6121, 6126, 6135, 6136, 6144, 6147, 6148, 6163, 6169, 6175, 6190, 6201, 6204, 6213, 6220, 6231, 6235, 6236, 6238, 6268, 6280, 6285, 6294, 6308, 6309, 6330, 6345, 6361, 6366, 6371, 6373, 6385, 6393, 6397, 6406, 6421, 6424, 6445, 6448, 6469, 6488, 6490, 6493, 6495, 6504, 6525, 6534, 6535, 6538, 6540, 6543, 6546, 6553, 6555, 6558, 6559, 6561, 6576, 6588, 6595, 6604, 6616, 6628, 6630, 6633, 6634, 6646, 6650, 6663, 6664, 6669, 6678, 6693, 6708, 6709, 6714, 6718, 6721, 6730, 6731, 6747, 6749, 6753, 6754, 6760, 6766, 6774, 6780, 6781, 6784, 6793, 6801, 6819, 6825, 6834, 6840, 6847, 6853, 6861, 6866, 6874, 6875, 6879, 6880, 6886, 6889, 6938, 6945, 6949, 6951, 6961, 6965, 6972, 6979, 6990, 6994, 7006, 7008, 7009, 7015, 7018, 7019, 7020, 7035, 7053, 7056, 7071, 7072, 7078, 7104, 7105, 7109, 7110, 7128, 7138, 7141, 7150, 7164, 7170, 7180, 7183, 7198, 7200, 7204, 7206, 7239, 7240, 7244, 7246, 7248, 7255, 7258, 7260, 7263, 7271, 7288, 7293, 7309, 7316, 7320, 7323, 7325, 7326, 7330, 7338, 7359, 7360, 7369, 7372, 7375, 7379, 7381, 7413, 7416, 7429, 7449, 7455, 7456, 7461, 7470, 7473, 7495, 7501...


In bold are double peloton pairs.

Note that three triplets also emerged:

I imagine there are many more of these...

Rohan Lewis

2023.09.22

Code can be found here.