Math Investigation 6g602v

COMPUTER INVESTIGATION 1.3 Palindromic Sums 1. Yes 2. The following list contains all the two-digit numbers that require two or more steps. 2 steps: 19, 28, 37, 39, 46, 48, 49, 55, 57, 58, 64, 66, 67, 73, 75, 76, 82, 84, 85, 91, 93, and 94 3 steps: 59, 68, 77, 86, and 95 4 steps: 69, 78, 87, and 96 6 steps: 79, 88, 97, and 99 24 steps: 89 and 98 3. Yes, for two-digit numbers that go to palindromic numbers in 2 steps, the sums of their digits are either 10, 12, or 13. After the first four of these numbers (19, 28, 37, 39), the sums of 10, 12, and 13 repeat for the remaining two-digit numbers in the above list. All two-digit numbers that require 3 steps have a digit sum of 14; for 4 steps the sum is 15; for 6 steps the sums are 16 and 18; and for 24 steps the sum is 17. The sums of the digits of two-digit numbers range from 1 to 18. For a two digit number ab with a + b ≤ 9 or a + b = 11, the number goes to a palindrome in one step, as shown by the following equations. a(10) + b + b(10) + a (a+b)10 + (a+b) = 10a + 10b + a + b = 11a + 11b = 11(a + b) For a + b = k with k ≤ 9, then ab goes to the palindromic number 11k in one step. If a + b = 11, then ab goes to the palindrome 11(11) = 121 in one step. 4,5. There are only eight three-digit numbers that require exactly 23 steps to go to palindromic numbers. They are listed here. The sum of the digits for each of these numbers is 16. 187 286 385 484 583 682 781 880 All other three-digit numbers except the following 13 require less than 23 steps. (Note: The Mathematics Investigator software is programmed to stop the process of reversing and adding numbers when sums are reached that have over 100 digits.) 196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986 The sums of the digits in each of these 13 numbers are either 16, 23, or 24. However, there are other threedigit numbers whose digit sum is 16, 23, or 24 which do go to palindromic numbers. The 13 numbers that do not go to palindromic numbers in 23 or fewer steps can also be classified by looking at the middle (tens) digit of the number and the outer two digits: those with a middle digit of 9 and outer digits whose sum is 7; those with a middle digit of 8 and outer digits whose sum is 15; and those with a middle digit of 7 and outer digits whose sum is 17. The only three-digit numbers that satisfy these conditions are the 13 listed above. 6. There are 236 four-digit numbers that do not go to palindromic numbers in fewer than 220 steps. These numbers areshown in increasing order in the following list. There are many patterns in these numbers. Notice that in the first few rows the differences between consecutive numbers alternate between 2 and 88 with the exception of a few numbers. The digits of the numbers in the top row of the table also follow another pattern: the inner two digits of each number have a sum of 13 and the sum of the two outer digits alternates between 6 and 8. The sum of the inner digits of the first number in the second row (1997) is 18 and the sum of the two outer digits is 8. The next 12 numbers follow the pattern of the numbers in the first row of the table, and then there is another number (2996) whose inner digit sum is 18 and whose outer digit

sum is 8. These first few rows suggest that we look at the sums of inner digits and the sums of outer digits for the 236 numbers. 1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997, 2494, 2496, 2584, 2586, 2674, 2676, 2764, 2766, 2854, 2856, 2944, 2946, 2996, 3493, 3495, 3583, 3585, 3673, 3675, 3763, 3765, 3853, 3855, 3943, 3945, 3995, 4079, 4169, 4259, 4349, 4439, 4492, 4494, 4529, 4582, 4584, 4619, 4672, 4674, 4709, 4762, 4764, 4799, 4852, 4854, 4889, 4942, 4944, 4979, 4994, 5078, 5168, 5258, 5348, 5438, 5491, 5493, 5528, 5581, 5583, 5618, 5671, 5673, 5708, 5761, 5763, 5798, 5851, 5853, 5888, 5941, 5943, 5978, 5993, 6077, 6167, 6257, 6347, 6437, 6490, 6492, 6527, 6580, 6582, 6617, 6670, 6672, 6707, 6760, 6762, 6797, 6850, 6852, 6887, 6940, 6942, 6977, 6992, 7059, 7076, 7149, 7166, 7239, 7256, 7329, 7346, 7419, 7436, 7491, 7509, 7526, 7581, 7599, 7616, 7671, 7689, 7706, 7761, 7779, 7796, 7851, 7869, 7886, 7941, 7959, 7976, 7991, 8058, 8075, 8079, 8089, 8148, 8165, 8169, 8179, 8238, 8255, 8259, 8269, 8328, 8345, 8349, 8359, 8418, 8435, 8439, 8449, 8490, 8508, 8525, 8529, 8539, 8580, 8598, 8615, 8619, 8629, 8670, 8688, 8705, 8709, 8719, 8760, 8778, 8795, 8799, 8809, 8850, 8868, 8885, 8889, 8899, 8940, 8958, 8975, 8979, 8989, 8990, 9057, 9074, 9078, 9088, 9147, 9164, 9168, 9178, 9237, 9254, 9258, 9268, 9327, 9344, 9348, 9358, 9417, 9434, 9438, 9448, 9507, 9524, 9528, 9538, 9597, 9614, 9618, 9628, 9687, 9704, 9708, 9718, 9777, 9794, 9798, 9808, 9867, 9884, 9888, 9898, 9957, 9974, 9978, 9988, 9999 The following table contains the classification of the numbers from the above list according to the sums ofIt is natural to wonder if inner digit sums and outer digit sums can be used to classify five-digit numbers. This may be true. Consider the following patterns. All numbers such as 99883, 38989, 58897, 67996, 99973, etc., whose inner three digit sum is 25, whose outer two digit sum is 12, and which do not have 7 as the middle (hundreds) digit, do not go to palindromic numbers in less than 220 steps. There are 35 four-digit numbers that satisfy the preceding conditions. All four-digit numbers whose inner three digit sum is 25, whose outer two-digit sum is 12, and whose middle digit is 7, such as 99793, 89794, 59797, etc., go to a palindromic number in five steps. You may wish to pursue this investigation further. It appears to be unexplored territory. So far it has been determined that: all two-digit numbers go to Palindromic numbers; all but 1.444...% of the three-digit numbers go to palindromic numbers; all but 2.6222...% of the four-digit numbers go to palindromic numbers; and all but 6.49111...% of the five-digit numbers go to palindromic numbersthe inner and outer digits. These numbers fall into 12 classes, and every four-digit number that has the inner and outer digit sums shown in this table is also in the above list of 236 numbers. Sums of Inner Digits Sums of Outer Digits Number of Four-digit Numbers 18 18 1 18 8 8 16 13 18 16 17 6 17 17 4 14 16 15 13 6 36 13 8 48 8 17 18 7 17 16 7 13 48 5 16 18 Total 236