Problem #1: A number is an anti-Friedman number if it has no repeated digits and it can be formed using one of each digit NOT in the number, together with addition, subtraction, multiplication, division, exponentiation, and concatenation. For example, 592710 = 843 + 6. There are finitely many anti-Friedman numbers, and it might be possible to find them all. What is the largest one you can find? What are the smallest numbers that are NOT anti-Friedman numbers?
Problem #2: A number is a k-shifted Friedman number if it can be written using its digits shifted by some constant k. The constant k can be positive or negative as long as the shifted digits are all between 0 and 9. For example, with k=1, 108 = 12 × 9. If the shifted digits can be used in order, we call the number a nice k-shifted Friedman number. For example, with k=1, 178 = 2 × 89. What are the small k-shifted Friedman numbers? Which of them are nice?
Problem #3: A pair of numbers is a Friedman pair if the digits of the first can make the second, and the digits of the second can make the first. For example, 27 = 28 – 1 and 128 = 27. If the digits can be used in order, we call the pair a nice Friedman pair. If only one of them can be written with the digits in order, we call it a semi-nice Friedman pair. What are the small Friedman pairs? Which of them are semi-nice or nice? We can also ask for Friedman triplets, 3 numbers which can be formed by the digits of either of the other two.
Problem #1:
Marc Lapierre confirmed that all numbers with 3 or fewer digits are anti-Friedman numbers. Are all numbers with 4 digits anti-Friedman numbers?
Here are the known Anti-Friedman numbers with 5 or more digits:
| 10246 = 75 – (9 / 3)8 | 10273 = 64 × 8 – 95 | 10275 = 64 × 8 – 93 | 10276 = (83 × 5 + 9) × 4 | 10293 = 64 × 8 – 75 |
| 10295 = 64 × 8 – 73 | 10352 = 64 × 8 – 9 – 7 | 10357 = 64 × 8 – 9 – 2 | 10359 = 64 × 8 – 7 – 2 | 10368 = 45 × 9 + 27 |
| 10372 = 64 × 8 + 9 – 5 | 10375 = 64 × 8 + 9 – 2 | 10379 = (64 + 2) × 8 – 5 | 10395 = 64 × 8 + 27 | 10467 = (592 + 8) × 3 |
| 10496 = 27 × (85 – 3) | 10584 = 63 × 7 × (9 – 2) | 10632 = 54 × (9 + 8) + 7 | 10648 = (2 + 9) × (7 – 5)3 | 10657 = (8 × 2)3 + 94 |
| 10752 = 896 × 4 × 3 | 10764 = (53 – 8) × 92 | 10792 = 64 × 8 + 53 | 10845 = (9 + 6) × 723 | 10923 = (48 + 7 – 5) / 6 |
| 10925 = (46 × 8 + 7) / 3 | 10943 = (6 / 2)7 × 5 + 8 | 10952 = (64 + 73) × 8 | 10956 = 372 × 8 + 4 | 10976 = (2 + 5)3 × 8 × 4 |
| 12036 = (74 + 8) × 5 – 9 | 12039 = (74 + 8) × 5 – 6 | 12096 = 7 × (4 × 3)8–5 | 12098 = 6 × 37 – 45 | 12309 = (46 + 7) × (8 – 5) |
| 12780 = 639 × 5 × 4 | 13056 = 27 × (98 + 4) | 13092 = (48 – 76) / 5 | 13098 = ((5 – 2)7 – 4) × 6 | 14580 = 93 × (2 × 7 + 6) |
| 14760 = 328 × 9 × 5 | 14820 = 76 × 39 × 5 | 15092 = 73 × (8 × 6 – 4) | 15628 = 3907 × 4 | 15678 = 402 × 39 |
| 15709 = (2 + 3)6 + 84 | 16038 = 297 × 54 | 16082 = 75 – 93 + 4 | 16093 = (482 – 5) × 7 | 16308 = 47 – 92 + 5 |
| 16349 = (85 – 70) / 2 | 16350 = 47 – 2 × (8 + 9) | 16375 = 4(8–20) – 9 | 16379 = 4(8–20) – 5 | 16384 = (9 – 5)7 + 2 × 0 |
| 16385 = 47 + 290 | 16389 = 47 + 5 + 2 × 0 | 16390 = 47 + 2 × (8 – 5) | 16392 = 47 + 8 + 5 × 0 | 16407 = (38 × 5 + 9) / 2 |
| 16704 = 58 × 32 × 9 | 16802 = 493 / 7 – 5 | 16804 = 75 – 4 + 3 × 2 | 16805 = 493 / 7 – 2 | 16807 = (9 + 4 – 2 × 3)5 |
| 16809 = 75 – 9 + 3 × 2 | 16830 = 75 + 92 / 4 | 16840 = 75 + (2 + 9) × 3 | 16890 = 75 + 34 + 2 | 16903 = 75 + 2 × 48 |
| 17082 = 5694 × 3 | 17280 = 43 × 9 × 6 × 5 | 17490 = 32+5 × 8 – 6 | 17496 = 8 × 32+5 + 0 | 17502 = 94 × 8 / 3 + 6 |
| 17504 = 92 × 63 + 8 | 17508 = (93 × 4 + 2) × 6 | 17560 = ((9 + 4)3 – 2) × 8 | 17580 = 269/3 + 4 | 17640 = (35 + 2) × 8 × 9 |
| 17820 = 396 × 45 | 17920 = 64 × 35 × 8 | 18026 = 59–3 + 74 | 18029 = 56 + 74 + 3 | 18270 = (45 – 9) × 6 × 3 |
| 18306 = (45 – 7) × 9 × 2 | 18504 = (73 × 6 – 2) × 9 | 18540 = (73 × 6 + 2) × 9 | 18720 = 936 × 5 × 4 | 18750 = 6 × (3 + 2)9–4 |
| 19084 = 367 × 52 | 19205 = 8 × 74 – 6 + 3 | 19206 = 8 × 74 – 5 + 3 | 19208 = (56 / 4)3 × 7 | 19240 = 65 × 37 × 8 |
| 19682 = 34+5 – 70 | 19683 = (2 + 70)4+5 | 19684 = 273 + 50 | 20169 = 35 × (87 – 4) | 20416 = (73 + 9) × 58 |
| 20451 = 39 + 768 | 20457 = 6819 × 3 | 20574 = 381 × 9 × 6 | 20731 = (8 × 9 / 6)4 – 5 | 20735 = (8 × 9 / 6)4 – 1 |
| 20736 = (8 + 4)9–5 × 1 | 20754 = 6918 × 3 | 21450 = (83 × 7 – 9) × 6 | 21504 = 89/3 × 7 × 6 | 21609 = 74 × 3 × (8 – 5) |
| 21658 = 3094 × 7 | 21870 = 34 × 9 × 6 × 5 | 21950 = (7 × 4)3 – 8 + 6 | 21958 = (7 × 4)3 + 6 + 0 | 21970 = (8 + 5)3 × (6 + 4) |
| 24056 = 97 × 31 × 8 | 24507 = 8169 × 3 | 24701 = (6 + 8)3 × 9 + 5 | 24705 = ((6 + 8)3 + 1) × 9 | 25803 = 61 × 47 × 9 |
| 25960 = (38 – 71) × 4 | 26039 = (57 – 8) / (4 – 1) | 26048 = (57 + 19) / 3 | 26180 = 935 × 7 × 4 | 26487 = 109 × 35 |
| 26510 = (74 + 9) × (8 + 3) | 26910 = 345 × 78 | 27504 = 9168 × 3 | 27810 = (45 + 6) × 9 × 3 | 28035 = (46 – 91) × 7 |
| 28156 = 7039 × 4 | 28507 = 134 – 9 × 6 | 28560 = (9 + 7 – 3)4 – 1 | 28561 = (7 + 9 – 3)4 + 0 | 28570 = (16 – 3)4 + 9 |
| 28591 = (7 + 6)4 + 30 | 28651 = 4093 × 7 | 29104 = (36 × 5 – 7) × 8 | 30296 = 541 × 8 × 7 | 30576 = 91 × 42 × 8 |
| 30618 = 92 × 54 × 7 | 30712 = (65 – 98) × 4 | 30752 = 961 × 8 × 4 | 31074 = ((8 × 9)2 – 5) × 6 | 31087 = (65 – 2) × 4 – 9 |
| 31089 = (65 – 2) × 4 – 7 | 31094 = (56 – 78) × 2 | 31408 = (56 + 79) × 2 | 31920 = 84 × 76 × 5 | 32018 = 47 + 56 + 9 |
| 32701 = (46 – 9) × 8 + 5 | 32710 = 85 – 9 × 6 – 4 | 32714 = 85 – 9 × 6 + 0 | 32760 = 85 + 14 – 9 | 32768 = (9 – 1)5 + 4 × 0 |
| 32769 = 85 + 140 | 32780 = (94 – 6 + 1) × 5 | 32805 = (6 + 9) × (4 – 1)7 | 32806 = 5 × 94 + 17 | 32810 = (94 + 7 – 6) × 5 |
| 32890 = 715 × 46 | 34902 = 5817 × 6 | 35904 = (672 – 1) × 8 | 36015 = 492 × (8 + 7) | 36508 = 9127 × 4 |
| 37908 = 52 × (4 – 1)6 | 38410 = (2 × 7)9–5 – 6 | 38416 = (2 × 7)9–5 + 0 | 39062 = (57 – 1) × 4 / 8 | 41067 = ((5 + 8) × 9)2 × 3 |
| 41503 = (69 + 8)2 × 7 | 43681 = 2097–5 | 45801 = (36 – 2) × 7 × 9 | 45901 = 38 × 7 – 26 | 45906 = (38 – 1 – 2) × 7 |
| 45920 = 7 × 38 – 6 – 1 | 45921 = 7 × 38 – 6 + 0 | 45926 = 7 × 38 – 1 + 0 | 45927 = 7 × 38 + 1 × 0 | 46095 = (37 + 8) × 21 |
| 47089 = (651 / 3)2 | 47320 = 91 × 65 × 8 | 48015 = (732 + 6) × 9 | 49170 = (28+5 + 3) × 6 | 50176 = (4 × 2)3 × 98 |
| 50421 = 78–3 × (9 – 6) | 50429 = 76–1 × 3 + 8 | 50617 = (2 × 9 – 3)4 – 8 | 50618 = (2 × 9 – 3)4 – 7 | 50619 = (8 + 7)4 – 2 × 3 |
| 50628 = (9 + 7 – 1)4 + 3 | 50629 = (8 + 7)4 + 3 + 1 | 50784 = 6 × (93 – 1)2 | 51840 = (36 – 9) × 72 | 51870 = (932 – 4) × 6 |
| 51980 = (76 × 3)2 – 4 | 51984 = (76 × 3)2 + 0 | 58401 = 927 × 63 | 59041 = (6 + 3)7–2 – 8 | 59042 = 81 × 36 – 7 |
| 59043 = (8 + 1)7–2 – 6 | 59046 = (8 + 1)7–2 – 3 | 59047 = 81 × 36 – 2 | 59048 = 37+6/2 – 1 | 60418 = 95 + 372 |
| 61054 = 73 × 89 × 2 | 65031 = 48 – 29 + 7 | 65128 = 9304 × 7 | 65821 = 9403 × 7 | 67081 = ((9 – 5)4 + 3)2 |
| 69120 = (8 × 3)7–4 × 5 | 71680 = 29 × 35 × 4 | 72105 = (94 – 6) × (8 + 3) | 74082 = (51 – 9)3 – 6 | 74086 = (51 – 9)3 – 2 |
| 74089 = ((5 + 2) × 6)3 + 1 | 76830 = ((9 + 5)4 – 1) × 2 | 78061 = 59–2 – 43 | 78125 = (9 – 4)(6+30) | 78126 = 54+3 + 90 |
| 78129 = 5(6+30) + 4 | 78134 = 5(6+20) + 9 | 79508 = 436/2 + 1 | 80352 = 64 × (71 – 9) | 80659 = (71 × 4)2 + 3 |
| 81902 = 47 × 5 – 6 × 3 | 81920 = (7 – 3)6 × 5 × 4 | 81950 = (47 + 23) × 5 | 81960 = (47 + 6) × (2 + 3) | 83509 = 174 – 6 × 2 |
| 83521 = (9 + 7 + 60)4 | 92160 = 45 × (87 + 3) | 93750 = 6 × (4 + 1)8–2 | 96032 = (5 × 74 – 1) × 8 | 98260 = 4 × 5 × 173 |
| 98301 = 6 × 47 – 5 + 2 | 98304 = (5 + 1) × (6 – 2)7 | 98305 = 6 × 47 + 2 – 1 | 98306 = (5 + 1) × 47 + 2 | 201684 = (3 + 9) × 75 |
| 327680 = 5 × 49–1 | 390617 = 58 – 2 × 4 | 390618 = 52×4 – 7 | 390625 = (17 + 4)8 | 390642 = 58 + 17 |
| 592710 = 843 + 6 |
Here are the smallest known ordinary shifted Friedman numbers, usually those with with 4 or fewer digits, and the nice shifted Friedman numbers with 5 or fewer digits:
| 99999988 = 1010–1 – 11 × 1 (JD) | 99999989 = 1011–1–1 – 11 | 999999988 = 1010–1 – 11 – 1 × 1 |
| 999999989 = 1011–1–1 – 11 × 1 | 999999998 = 1011–1–1 – 1 × 1 – 1 (JD) | 999999999 = (11 – 1)11–1–1 – 1 × 1 (JD) |
| 99999988 = (12 – 2)2×2×2 – 12 (JD) | 99999998 = (12 – 2)2+2+2+2 – 2 (JD) | 9999999998 = (2 × 2 × 2 + 2)2×2×2+2 – 2 × 1 | 9999999999 = (2 × 2 × 2 + 2)2×2×2+2 – 2 / 2 |
| 999997 = (3 × 3 + 1)3+3 – 3 (GA) | 999999998 = (3 × 3 + 3 / 3)3×3 + 3 – 3 – 2 (GA) |
| 99999988 = (4 + 3 + 3)4+4 – 4 – 4 – 4 | 99999997 = (4 + 4 + 2)4+4 – 4 + 4 / 4 | 99999999 = ( (44 – 4) / 4)4+4 – 4 / 4 |
| 7776 = (3 + 3)3+2 | 7779 = 3 + (3 + 3)5 | 99995 = –5 + (5 + 5)5 × 1 | 99999 = (5 + 5)5 – 5 / 5 |
| 6559 = 32+6 – 2 | 6859 = (52 – 6)3 | 7569 = (34 + 6)2 | 46656 = (1 × 3 + 3)2×3 | 46657 = 1 + (3 + 3)2+4 |
| 46659 = 1 × 3 + (3 × 2)6 | 46669 = 13 + (3 + 3)6 | 46875 = 1 × 3 × 54+2 | 46899 = 1 × 35 + 66 | 69984 = (3 + 6) × 56 × 1 |
| 49 = 72 | 648 = 64 / 2 | 3375 = 1 × 153 | 3844 = 1 × 622 | 3997 = 571 × 7 |
| 6478 = 5 × 64 – 2 | 6557 = 35+3 – 4 | 6558 = (3 + 6)4 – 3 | 6565 = (3 × 3)4 + 4 | 6859 = (76 / 4)3 |
| 7776 = (5 + 5 – 4)5 | 8464 = (46 × 2)2 | 36855 = (–1 + 46) × 3 × 3 | 36864 = ( (–1 + 4) × 64)2 | 38876 = (–1 + 6) × 65 – 4 |
| 42875 = ( (20 + 6) × 5)3 | 46648 = –2 × 4 + (4 + 2)6 | 46649 = (2 + 4)4+2 – 7 | 46688 = 2 × 4 × 4 + 66 | 65534 = 43×3–1 – 2 |
| 65536 = 43×3 × 1 / 4 | 87876 = 6 × (5 + (6 + 5)4) |
| 255 = –1 + 44 | 256 = (5 – 1)4 | 384 = 27 × 3 | 625 = 51×4 | 3584 = 7 × (2 × 4)3 |
| 3645 = 5 × 32+4 | 3647 = 2 + 5 × 36 | 3721 = 612 + 0 | 3722 = 612 + 1 | 3723 = 612 + 2 |
| 3724 = 612 + 3 | 3725 = 612 + 4 | 3726 = 612 + 5 | 3727 = 612 + 6 | 3728 = 612 + 7 |
| 3729 = 612 + 8 | 3837 = 622 – 7 | 4278 = 713 × 6 | 4374 = 3 × 2 × 36 | 4388 = (37 + 7) × 2 |
| 4588 = 74 + 37 | 5928 = 741 × 8 | 6556 = (5 + 4)4 – 5 | 7389 = 862 – 7 | 7689 = 65 – 87 |
| 7698 = 65 – 78 | 7776 = (6 + 6 – 6)5 | 7777 = (66 + 6) / 6 | 8754 = 4 × 37 + 6 | 16379 = 0 – 5 + 26+8 |
| 19683 = (0 + 8 – 5)7+2 | 24577 = 13 + 46 × 6 | 28556 = (17 – 4)4 – 5 | 32758 = 2 × (1 – 6 + 47) | 32768 = 2 × (–16 + 5)7 |
| 32896 = (2 × 1)7 + 85 | 46793 = 3 × 56 – 82 | 46868 = 3 × 57 / 5 – 7 | 58368 = 47–2 × 57 | 65531 = –5 + 44×2 + 0 |
| 65532 = –5 + 44×2 + 1 | 65533 = –5 + 44×2 + 2 | 65534 = –5 + 44×2 + 3 | 65535 = –5 + 44×2 + 4 | 65536 = –5 + 44×2 + 5 |
| 65537 = –5 + 44×2 + 6 | 65538 = –5 + 44×2 + 7 | 65539 = –5 + 44×2 + 8 | 82944 = (7 – 1) × (8 × 3)3 | 85536 = (7 + 4) × (4 + 2)5 |
| 99936 = –8 × 8 + (8 + 2)5 |
| 18 = 2 × 9 | 81 = 92 | 108 = 12 × 9 | 178 = 2 × 89 | 216 = 3 × 72 |
| 241 = 35 – 2 | 376 = 47 × 8 | 512 = (6 + 2)3 | 518 = 6 + 29 | 1008 = 112 × 9 |
| 1023 = 42+3 – 1 | 1024 = (2 + 3 – 1)5 | 1064 = 152 × 7 | 1152 = 32 × 62 | 1184 = 592 × 2 |
| 1248 = 39 × 25 | 1302 = 42 × 31 | 1458 = 6 × 95/2 | 1518 = 22 × 69 | 1521 = (62 + 3)2 |
| 1526 = 763 × 2 | 1652 = 236 × 7 | 1716 = 22 × 78 | 1720 = 123 – 8 | 1728 = 3 × 9 × 82 |
| 1736 = 248 × 7 | 1756 = (6 × 7)2 – 8 | 1778 = 2 × 889 | 2116 = (72 – 3)2 | 2163 = 37 – 24 |
| 2183 = 39–2 – 4 | 2187 = 98/2 / 3 | 2261 = 323 × 7 | 2365 = 74 – 36 | 2401 = (5 + 2)3+1 |
| 2406 = 73+1 + 5 | 2436 = 74 + 35 | 2532 = 633 × 4 | 2576 = 368 × 7 | 2601 = (3 × 17)2 |
| 2786 = 398 × 7 | 2874 = 958 × 3 | 3128 = 92 × 34 | 3136 = 4 × (4 × 7)2 | 3164 = 452 × 7 |
| 3168 = 792 × 4 | 3372 = 843 × 4 | 3402 = 14 × 35 | 3456 = (5 + 7)4 / 6 | 3481 = 594–2 |
| 3645 = 5 × (7 – 4)6 | 3804 = 951 × 4 | 3827 = 89 × 43 | 3852 = 4 × 963 | 4067 = 581 × 7 |
| 4087 = 85–1 – 9 | 4101 = 5 + 212 | 4176 = 72 × 58 | 4361 = (54 – 2) × 7 | 4367 = 54 × 7 – 8 |
| 4376 = 547 × 8 | 4506 = 751 × 6 | 4580 = 916 × 5 | 4655 = 665 × 7 | 4805 = 961 × 5 |
| 4865 = 695 × 7 | 5178 = (8 × 9)2 – 6 | 5324 = 4 × (6 + 5)3 | 5748 = 958 × 6 | 5832 = 94 – 36 |
| 5856 = 976 × 6 | 6176 = 772 × 8 | 6638 = 94 + 77 | 6776 = 77 × 88 | 7203 = 3 × (8 – 1)4 |
| 8183 = 29+4 – 9 | 8201 = 213 + 9 | 8836 = 949–7 | 12167 = 232–7+8 | 13822 = –2 + (4 × (9 – 3) )3 |
| 14336 = 25+4 × 4 × 7 | 14346 = 2 × (5 + 45 × 7) | 15540 = 2 × (–6 + 65) × 1 | 15550 = 2 × (66 / 6 – 1) | 15552 = –2 + (66 + 6) / 3 |
| 15554 = 2 × (6 / 6 + 65) | 15625 = (26 – 7 × 3)6 | 16367 = –2 – 7 + 47 – 8 | 16370 = –2 × 7 + 48–1 | 16384 = 27×4–9–5 |
| 16462 = –2 + 75 – 73 | 16536 = 2 × 76 + 47 | 17064 = 28 + 1 + 75 | 17152 = 28 × (26 + 3) | 17778 = 2 × 8889 |
| 18225 = (–2 + 9 × 3) × 36 | 20733 = –3 + (1 × 8 + 4)4 | 25281 = –3 + (6 × 3 × 9)2 | 25466 = (36 × 5 – 7) × 7 | 27648 = (–3 + 8 + 7)5 / 9 |
| 27702 = 38× (8 + 1)3 | 28561 = (3 × 9 × 6 + 7)2 | 31104 = 4 × (2 × (2 + 1) )5 | 32704 = –43 + 81×5 | 32761 = –4 – 3 + 87–2 |
| 32772 = 4 + (3 × 8 + 8)3 | 35152 = (–4 + 6) × 263 | 35721 = ( (–4 + 6) × 8 + 3)2 | 36828 = (–4 + (7 + 9)3) × 9 | 38420 = 4 + (9 + 5)3+1 |
| 45355 = – 5 + 64 + 66 | 45360 = 5 × 64 × 7 × 1 | 47223 = 583 × 34 | 50420 = (6 + 1)5 × 3 – 1 | 54432 = 65 × (– 5 + 4 × 3) |
| 54463 = (65 + 5) × 7 – 4 | 54516 = (65 + 6 × 2) × 7 | 54872 = (6 + (–5 + 9) × 8)3 | 65537 = 76–6 + 48 | 73728 = 8 × 48–3 × 9 |
| 73776 = 848 × 87 | 78036 = –89 + (1 + 4)7 | 78126 = –8 + 9 + (2 + 3)7 | 84402 = 9 × (55 + 1) × 3 | 86436 = 9 × 75 × 4 / 7 |
| 88210 = (99 × 3)2 + 1 | 88211 = (99 × 3)2 + 2 | 88212 = (99 × 3)2 + 3 | 88213 = (99 × 3)2 + 4 | 88214 = (99 × 3)2 + 5 |
| 88215 = (99 × 3)2 + 6 | 88216 = (99 × 3)2 + 7 | 88217 = (99 × 3)2 + 8 | 88218 = (99 × 3)2 + 9 |
| 12 = 3 × 4 | 24 = 4 × 6 | 35 = 5 × 7 | 56 = 7 × 8 | 117 = 3 × 39 |
| 235 = 47 × 5 | 260 = 28 + 4 | 324 = 54 × 6 | 1020 = 322 – 4 | 1021 = 43+2 – 3 |
| 1023 = –3 + 2 + 45 | 1024 = 326–4 | 1026 = 48–3 + 2 | 1032 = 45 + 23 | 1167 = 3 × 389 |
| 1243 = 64 – 53 | 1352 = (73 – 5) × 4 | 1456 = (63 – 8) × 7 | 1470 = (93 + 6) × 2 | 1554 = 377 × 6 |
| 1600 = (38 + 2)2 | 1603 = (8 × 5)2 + 3 | 1664 = (63 – 8) × 8 | 1764 = 3 × 98 × 6 | 2043 = 46 / 2 – 5 |
| 2050 = 24+7 + 2 | 2051 = 24+7 + 3 | 2052 = 24+7 + 4 | 2053 = 24+7 + 5 | 2054 = 24+7 + 6 |
| 2055 = 24+7 + 7 | 2056 = 24+7 + 8 | 2057 = 24+7 + 9 | 2064 = 86 × 24 | 2205 = 472 – 4 |
| 2235 = 447 × 5 | 2256 = 48 × 47 | 2304 = (54 – 6)2 | 2345 = 74 – 56 | 2401 = (46 + 3)2 |
| 2405 = 74 + 6 – 2 | 2407 = 492 + 6 | 2425 = 74 + 6 × 4 | 2601 = (48 + 3)2 | 3025 = (5 × (4 + 7))2 |
| 3053 = 55 – 72 | 3125 = (3 × 4 – 7)5 | 3130 = 55 + 3 + 2 | 3131 = 55 + 3 + 3 | 3132 = 55 + 3 + 4 |
| 3133 = 5 + 3 + 55 | 3134 = 55 + 3 + 6 | 3135 = 55 + 3 + 7 | 3136 = 55 + 3 + 8 | 3137 = 55 + 3 + 9 |
| 3143 = 55 + 6 × 3 | 3163 = 55 + 38 | 3240 = 64 × 5 / 2 | 3275 = (47 – 9) / 5 | 3324 = 554 × 6 |
| 3341 = 55 + 63 | 3375 = 75 × 9 × 5 | 3525 = 75 × 47 | 3600 = (58 + 2)2 | 3672 = 459 × 8 |
| 4102 = 46 + 3 × 2 | 4350 = 725 × 6 | 4536 = 567 × 8 | 4620 = 682 – 4 | 4624 = 686–4 |
| 4770 = 692 + 9 | 6075 = 782 – 9 | 6144 = 83 × (6 + 6) | 6147 = 683 × 9 | 6174 = 98 × 63 |
| 6561 = 38 × (8 – 7) | 7056 = 98 × 72 | 7560 = 872 – 9 | 7632 = 954 × 8 | 10000 = ( (3 + 2) × 2)2+2 |
| 10032 = 32 + (2 × 5)4 | 10131 = 3 × (2 + (3 × 5)3) | 10240 = (3 + 2 × 4)6 / 2 | 10750 = 3 × 29 × 7 – 2 | 11667 = 3 × 3889 |
| 12423 = 3 × (46 + 45) | 12456 = 3 × (46 + 7 × 8) | 12503 = 3 + 4 × (7 – 2)5 | 13116 = (–3 + 5) × (–3 + 38) | 13122 = (–3 + 5) × 34+4 |
| 14337 = (–3 + 6)5 × 59 | 15645 = 37 + 8 × 6 × 7 | 16134 = –3 + 83 + 56 | 16464 = 3 × 8 × 686 | 17514 = 3 × 973 × 6 |
| 20412 = 42 × 6 × 34 | 20732 = –4 + (–2 + 9 + 5)4 | 23120 = 4 × 5 × 342 | 23704 = 4 × 5926 | 24477 = 46 × 6 – 99 |
| 24576 = 46 × (7 – 9 + 8) | 25600 = 4 × (78 + 2)2 | 26170 = 48 – 39 × 2 | 31250 = 53–4+7 × 2 | 33324 = 5554 × 6 |
| 46647 = (6 + 8 – 8)6 – 9 | 46752 = 6 × 8 × 974 | 50632 = 7 + (2 + 8 + 5)4 | 54320 = 7 × (65 – 42) | 54355 = 7 × 65 – 77 |
| 54367 = 7 × (65 – 8) – 9 | 54425 = 7 × 6 × 64 – 7 | 54432 = 7 × 66–5+4 | 55741 = 7 × 7963 | 60516 = 82 × 738 |
| 62210 = (8 + 4)4 × 3 + 2 | 62211 = (8 + 4)4 × 3 + 3 | 62212 = (8 + 4)4 × 3 + 4 | 62213 = (8 + 4)4 × 3 + 5 | 62214 = (8 + 4)4 × 3 + 6 |
| 62215 = (8 + 4)4 × 3 + 7 | 62216 = (8 + 4)4 × 3 + 8 | 62217 = (8 + 4)4 × 3 + 9 | 64672 = 86 × 8 × 94 | 65536 = ( (– 8 + 7)7 + 5)8 |
| 66032 = 8 × 8254 |
| 342 = 57 × 6 | 1000 = (4 + 3 + 3)3 | 1003 = (4 + 6)3 + 3 | 1004 = (3 + 7)3 + 4 | 1012 = –4 × 3 + 45 |
| 1022 = 45 – 5 + 3 | 1024 = 43+7–5 | 1260 = 35 × 9 × 4 | 2106 = 54 × 39 | 2106 = 54 × 39 |
| 2401 = 75+3–4 | 2404 = 75 / 7 + 3 | 2431 = 74 + 6 × 5 | 2530 = (83 – 6) × 5 | 2560 = 5 × 89/3 |
| 2605 = 5 × (83 + 9) | 3504 = 73 × 8 × 6 | 4145 = 84 + 7 × 7 | 4201 = (75 – 3) / 4 | 5120 = 5 × 48–3 |
| 6502 = 38 – 59 | 6513 = 94 – 8 × 6 | 6521 = 94 – 8 × 5 | 6552 = (8 – 5)8 – 9 | 6553 = (9 – 6)8 – 8 |
| 6560 = 38 – 9 / 9 | 6561 = 98 / 94 | 10234 = (4 + 35 × 6) × 7 | 12360 = (45 + 6) × (9 + 3) | 14641 = (4 + 7)(9+7)/4 |
| 15613 = –4 – 8 + (9 – 4)6 | 20510 = 5 × (3 + 84 + 3) | 22064 = (55 + 3 × 9) × 7 | 22344 = (55 + 67) × 7 | 24000 = (5 × (7 – 3) )3 × 3 |
| 31220 = (6 + 4) × (55 – 3) | 31250 = (6 + 4) × 58–3 | 33152 = 6 × 64 + 85 | 35034 = 6 × (8 × 36 + 7) | 46151 = 7 × (94 + 8 × 4) |
| 46305 = 7 × (9 × 6 + 38) | 50420 = 8 + 3 × (75 – 3) | 50423 = 8 + 3 × 75 – 6 | 50616 = (8 × 3 – 9)4 – 9 | 54432 = (8 – 7) × 7 × 65 |
| 60502 = 93 × 83 – 5 | 64512 = 9 × 7 × (8 – 4)5 | 65015 = –9 – 83 + 48 |
| 120 = 5 × 6 × 4 | 125 = 59–6 | 315 = 7 × 5 × 9 | 513 = 9 × 57 | 1023 = 45 – 7 + 6 |
| 1024 = 85×4/6 | 1100 = 5 × 5 × 44 | 1120 = 5 × 56 × 4 | 1215 = 5 × (9 – 6)5 | 1235 = 95 × (6 + 7) |
| 1445 = 85 × (8 + 9) | 2305 = 74 – 96 | 2401 = (56 / 8)4 | 2403 = 74 + 8 – 6 | 3112 = –7 + 55 – 6 |
| 3140 = 785 × 4 | 3402 = 486 × 7 | 4051 = 84 – 9 × 5 | 4224 = 8 × 66 × 8 | 5313 = 759 × 7 |
| 5432 = 679 × 8 | 10000 = (54 – 44)4 | 10020 = 5 × 4 + (4 + 6)4 | 10240 = 5 × 46 / 8 × 4 | 11120 = 5 × 556 × 4 |
| 12500 = 56 / (9 – 4) × 4 | 15040 = 5 × 94 × 8 × 4 | 21120 = 6 × 55 × 64 | 22500 = 6 × 6 × (9 – 4)4 | 52110 = 965 × 54 |
| 52421 = – 9 + (6 + 86) / 5 |
| 210 = 7 × 6 × 5 | 1024 = (6 + 7 – 9)5 | 1032 = 86 × (5 + 7) | 1140 = 95 × (6 + 6) | 1304 = 69–5 + 8 |
| 2030 = 7 × 58 × 5 | 2304 = 9 × (7 – 5)8 | 4221 = 67 × 9 × 7 | 11204 = (–6 + 6 × 75) / 9 | 20000 = ( (7 – 5) × 5)5 / 5 |
| 20230 = 7 × 578 × 5 | 30200 = 8 × 5 × 755 | 31103 = (–8 + (6 + 6)5) / 8 |
| 103 = 97 + 6 | 112 = (7 + 7) × 8 | 1000000 = (7 + 6 / 6 + (6 + 6) / 6)6 (GA) |
| 1000000 = ((77 – 7) / 7)7+7–8 (JD) | 10000000 = (8 + 7 / 7 + 7 / 7)7 – 7 + 7 (GA) | 10000001 = (8 + 7 / 7 + 7 / 7)7 – 7 + 8 (GA) | 10000002 = (8 + 7 / 7 + 7 / 7)7 – 7 + 9 (GA) | 20000000 = (9 + 7 / 7)7 × (7 / 7 + 7 / 7) (GA) |
| 1000000 = (9 + 8 / 8)8–(8+8)/8 (JD) | 10000000 = (9 + 8 / 8)8–8/8 – 8 + 8 (JD) | 10000001 = (9 + 8 / 8)8–8/8 – 8 + 9 | |
| 100000000 = (9 + 8 / 8)8 + 8 × (8 + 8 – 8 – 8) | 100000001 = (9 + 8 / 8)8 + 8 × (8 – 8) – 8 + 9 | 100000010 = (9 + 8 / 8)8 + 8 + 8 / 8 + 9 – 8 | |
| 100000011 = (9 + 8 / 8)8 – 8 + 8 / 8 + 9 + 9 | 100000100 = (9 + 8 / 8)8 + 98 + (8 + 8) / 8 | 100000101 = (9 + 8 / 8)8 + 99 + (8 + 8) / 8 | 111111111 = ((9 + 9 / 9)9 – 9 / 9 – 9 + 9) / 9 (JD) |
Here are the known Friedman pairs, both of which have 3 or fewer digits, and the nice Friedman pairs, both of which have 4 or fewer digits.
| 53 = 51 + 2 125 = 53 | 27 = – 1 + 28 128 = 27 | 128 = 26+1 162 = 81 × 2 | 128 = 16 × 8 168 = 21 × 8 | 36 = 62×1 216 = 63 |
| 63 = 62 + 1 216 = 63 | 126 = 21 × 6 216 = 62+1 | 28 = – 2 + 5 × 6 256 = 28 | 192 = 25 × 6 256 = 29–1 | 216 = 65–2 256 = 162 |
| 144 = (7 + 5)2 257 = 1 + 44 | 174 = 2 × 87 287 = 41 × 7 | 128 = 32 × 4 324 = 182 | 184 = 23 × 8 328 = 41 × 8 | 37 = 34 + 3 343 = 73 |
| 175 = 35 × 5 355 = 71 × 5 | 243 = (1 + 2)5 512 = (2 × 4)3 | 546 = 91 × 6 619 = 54 – 6 | 54 = 56 – 2 625 = 54 | 243 = (6 / 2)5 625 = (3 + 2)4 |
| 364 = 52 × 7 725 = 36 – 4 | 136 = 27 + 8 728 = 36 – 1 | 36 = 7 + 29 729 = 36 | 63 = 72 – 9 729 = 36 | 126 = 7 × 2 × 9 729 = (1+2)6 |
| 137 = 27 + 9 729 = 37–1 | 194 = 97 × 2 729 = 94–1 | 243 = 27 × 9 729 = 34+2 | 256 = (9 + 7)2 729 = (–2 + 5)6 | 346 = 73 +3 733 = 36 + 4 |
| 625 = (10/2)4 1024 = (6–2)5 | 174 = 1 × 29 × 6 1296 = (–1+7)4 | 648 = 12 × 9 × 6 1296 = 6–4+8 | 1164 = 12 × 97 1297 = 1×1 + 64 | 263 = 1×7 + 28 1728 = (2×6)3 |
| 393 = 1 + 72 × 8 1728 = (3+9)3 | 256 = (–2+0+4)8 2048 = 25+6 | 1632 = 204 × 8 2048 = 163 / 2 | 2178 = (2+1)7 – 9 2179 = (2+1)7 – 8 | 1827 = 21 × 87 2187 = (18 + 2)7 |
| 502 = 2 + 500 2500 = 502 | 2736 = (2×7)3 – 8 2738 = (2×7)3 – 6 | 2735 = (2×7)3 – 9 2739 = (2×7)3 – 5 | 2536 = 29–1 × 6 2916 = (–1+5) × 36 | 1832 = 2 × 916 2916 = 183 / 2 |
| 1836 = 306 × 6 3066 = (–1 + 83) × 6 | 145 = (31–2)×5 3125 = (1+4)5 | 165 = (31+2) × 5 3125 = (–1+6)5 | 1536 = 3 × (3/6)–9 3369 = 153 – 6 | 2187 = 3–5+8+4 3584 = 21+8 × 7 |
| 2916 = 35 × (8+4) 3584 = 29 × (1+6) | 1935 = 3×645 3645 = 1 × 93 × 5 | 3645 = (–3+6)6 × 5 3665 = (36 + 4) × 5 | 3645 = (–3+8) × 93 3893 = 3 × 64 + 5 | 3648 = 38 × 96 3896 = 3 × 64 + 8 |
| 972 = 3 + 969 3969 = (9×7)2 | 2592 = 3 × 96 × 9 3969 = ((2+5) × 9)2 | 136 = 40 + 96 4096 = (1+3)6 | 354 = 40 × 9 – 6 4096 = (3+5)4 | 415 = 409 + 6 4096 = 41+5 |
| 1728 = (4×3)5–2 4352 = 17 × 28 | 2394 = – 4 – 3 + 74 4374 = 2 / 3 × 94 | 3182 = 43 × 74 4374 = 3–1+7 × 2 | 1692 = 47 × 62 4762 = 1 + 692 | 2744 = 4 × 7 × 98 4798 = 2 × 74 – 4 |
| 3125 = (4 + 80)5 4805 = 312 × 5 | 1944 = 486 × 4 4864 = 19 × 44 | 4284 = 51 × 84 5184 = 4 × (–2+8)4 | 6435 = (64 – 9) × 5 6495 = (64 + 3) × 5 | 4994 = – 6 + 54 × 8 6548 = – 4 – 9 + 94 |
| 3138 = 6 + 55 + 7 6557 = – 3 – 1 + 38 | 338 = 6 × 55 + 8 6558 = – 3 + 38 | 3835 = 65 × 59 6559 = 38 + 3 – 5 | 335 = 6 × 56 – 1 6561 = 33+5 | 337 = 6 × 56 + 1 6561 = 3 / 3–7 |
| 342 = 6 × (56+1) 6561 = 34×2 | 391 = 65 × 6 + 1 6561 = 39–1 | 3119 = – 6 + 56–1 6561 = 3–1+9×1 | 3965 = 65 × 61 6561 = 39–6+5 | 1294 = 65 / 6 – 2 6562 = – 1 + 2 + 94 |
| 3844 = (6+56)2 6562 = 38 + 4/4 | 394 = 65 × 6 + 4 6564 = 3 + 94 | 386 = – 6 + 56 × 7 6567 = 38 + 6 | 3438 = 6 × 573 6573 = 3×4 + 38 | 3882 = – 6 + 64 × 3 6643 = 38 + 82 |
| 3891 = (6 + 65) / 2 6652 = 38 + 91 | 2193 = 6 + (8–5)7 6857 = – 2 + 193 | 4656 = 776 × 6 7766 = – 4 + 65 – 6 | 5159 = 77 × 67 7767 = (5+1)5 – 9 | 1554 = 777 × 2 7772 = (1+5)5 – 4 |
| 2352 = – 7×7 + 74 7774 = (2×3)5 – 2 | 1715 = 7 × 7 × 7 × 5 7775 = – 1 + (7–1)5 | 245 = 7 × (– 7 + 7 × 6) 7776 = (2+4)5 | 1556 = 778 × 2 7782 = (1+5)5 + 6 | 6468 = 77 × 84 7784 = 64 × 6 + 8 |
| 6545 = 77 × 85 7785 = 65 + 4 + 5 | 1562 = 781 × 2 7812 = (– 1 + 56) / 2 | 6562 = 7 – 8 + 38 7838 = 65 + 62 | 1458 = 81 × 9 × 2 8192 = 1 × 45 × 8 | 2888 = 8 × 192 8192 = (2/8)–8 / 8 |
| 6553 = – 9 + 38 + 1 9381 = 6 + 55 × 3 | 2744 = (9+5)9–6 9596 = (– 2 + 74) × 4 | 4374 = 9 × 9 × 9 × 6 9996 = – 4 + (3+7)4 |
Alex Rower was interested in consecutive Friedman pairs:
| 2915 = (9 + 6)2 – 1 2916 = (9 × (5 + 1))2 (AR) | 4095 = 46 – 90 4096 = (9 – 50)4 (AR) | 4096 = (7 + 90)4 4097 = 46 + 90 (AR) | 6495 = 94 – 66 6496 = 94 – 65 (AR) | 9215 = 962 – 1 9216 = (95 + 1)2 (AR) |
| 9216 = (97 – 1)2 9217 = 962 + 1 (AR) |
Alex Rower was also interested in Friedman triple loops. Are there other 3-digit triples?
| 216 = 65–2 126 = 21 × 6 256 = 162 (AR) | 126 = 7 × 2 × 9 256 = 62+1 729 = (-2+5)6 (AR) |
If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 6/26/19.