# Problem of the Month (June 2019)

Several times, the problem of the month has been something to do with Friedman Numbers, whether it be S-Numbers, Self-Reproducing Numbers, or Fractional, Redundant, Almost, and Approximate Friedman Numbers. This month we consider a few other variations.

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.

Submissions were received from Joe DeVincentis, Alex Rower, Gordon Atkinson, and Marc Lapierre.

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:

Largest Anti-Friedman Numbers
 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

Problem #2:

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:

Shift of –8
 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)

Shift of –7
 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

Shift of –6
 999997 = (3 × 3 + 1)3+3 – 3 (GA) 999999998 = (3 × 3 + 3 / 3)3×3 + 3 – 3 – 2 (GA)

Shift of –5
 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

Shift of –4
 7776 = (3 + 3)3+2 7779 = 3 + (3 + 3)5 99995 = –5 + (5 + 5)5 × 1 99999 = (5 + 5)5 – 5 / 5

Shift of –3
 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

Shift of –2
 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)

Shift of –1
 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

Shift of +1
 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

Shift of +2
 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

Shift of +3
 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

Shift of +4
 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

Shift of +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

Shift of +6
 103 = 97 + 6 112 = (7 + 7) × 8 1000000 = (7 + 6 / 6 + (6 + 6) / 6)6 (GA)

Shift of +7
 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)

Shift of +8
 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)

Problem #3:

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.

Friedman Pairs
 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:

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?

Friedman Triple Loops
 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.