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Four fours is a classic problem, dating back at least 90 years. Create each of the positive integers using four copies of 4, and any standard operations. All four numbers must be used, but no others. Your solutions will be assigned an exquisiteness level.
Use the online submissions page to get your Integermania solutions posted here! This problem is now in semi-retired status, so you may submit an unlimited number of solutions each month.
Page 1 (1-400), Page 2 (401-800), Page 3 (801-1200), Page 4 (1201-1600), Page 5 (1601-2000), Page 6 (2001-2400), Page 7 (2401-2800), Page 8 (2801-3200), Page 9 (3201-3600), Page 10 (3601-4000), Page 11 (4001-4400), Page 12 (4401-4800), Page 13 (4801-16000), Page 41 (16001+).
| 4801 (4.8) $\cosh(4 \times \arsinh\sqrt{4!}) \times \dfrac44$ Steve Wilson, 6/25 Lawrence, KS |
4802 (4.8) $\cosh(4 \times \arsinh\sqrt{4!}) + \dfrac44$ Steve Wilson, 6/25 Lawrence, KS |
4803 (5.0) $\cosh(4 \times \arsinh\sqrt{4!})$ $\phantom. + \dfrac{4}{\sqrt{4}}$ Steve Wilson, 6/25 Lawrence, KS |
4804 (5.2) $\dfrac{4!}{(\cot\arctan\sqrt{4})\%}$ $\phantom. + \sqrt{4 \times 4}$ Steve Wilson, 6/25 Lawrence, KS |
4805 (5.0) $\cosh(4 \times \arsinh\sqrt{4!})$ $\phantom. + \sqrt{4 \times 4}$ Steve Wilson, 6/25 Lawrence, KS |
4806 (5.2) $\cosh(4 \times \arsinh\sqrt{4!})$ $\phantom. + \dfrac{\sqrt{4}}{.4}$ Steve Wilson, 6/25 Lawrence, KS |
4807 (5.0) $\cosh(4 \times \arsinh\sqrt{4!})$ $\phantom. + \dfrac{4!}{4}$ Steve Wilson, 6/25 Lawrence, KS |
4808 (5.0) $\dfrac{4!}{(\cot\arctan\sqrt{4})\%}$ $\phantom. + 4 + 4$ Steve Wilson, 6/25 Lawrence, KS |
4809 (4.8) $\cosh(4 \times \arsinh\sqrt{4!})$ $\phantom. + 4 + 4$ Steve Wilson, 6/25 Lawrence, KS |
4810 (4.0) $\dfrac{4 \times 4! + \sqrt{4\%}}{(\sqrt{4})\%}$ Steve Wilson, 6/25 Lawrence, KS |
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| 4817 (4.8) $\cosh(4 \times \arsinh\sqrt{4!})$ $\phantom. + 4 \times 4$ Steve Wilson, 6/25 Lawrence, KS |
4820 (3.8) $\dfrac{4 \times 4! + .4}{(\sqrt{4})\%}$ Steve Wilson, 6/25 Lawrence, KS |
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| 4830 (5.8) $\dfrac{4!}{(\cot\arctan\sqrt{4})\%}$ $\phantom. + \sqrt{\dfrac{4}{.\overline{4}\%}}$ Steve Wilson, 6/25 Lawrence, KS |
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| 4840 (4.8) $44 \times (\antilog\sqrt{4}$ $\phantom. + (\antilog 4)\pm)$ Steve Wilson, 6/25 Lawrence, KS |
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| 4845 (4.8) $\cosh(4 \times \arsinh\sqrt{4!})$ $\phantom. + 44$ Steve Wilson, 6/25 Lawrence, KS |
4850 (4.0) $\dfrac{\sqrt{4} - (4 + \sqrt{4})\%}{4\%\%}$ Steve Wilson, 6/25 Lawrence, KS |
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| 4860 (5.4) $\dfrac{4!}{(\cot\arctan\sqrt{4})\%}$ $\phantom. + \dfrac{4!}{.4}$ Steve Wilson, 6/25 Lawrence, KS |
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| 4863 (5.2) $\coth\ln\coth\arcosh(4! + 4!)$ $\phantom. + 4^4$ Steve Wilson, 4/25 Lawrence, KS |
4865 (5.2) $\coth\ln\coth\arsinh(4! + 4!)$ $\phantom. + 4^4$ Steve Wilson, 4/25 Lawrence, KS |
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| 4880 (3.8) $\dfrac{4 \times (4! + .4)}{(\sqrt{4})\%}$ Steve Wilson, 6/25 Lawrence, KS |
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| 4884 (5.4) $\cot\arctan\big(\sech\big(4 \times\phantom.$ $\arsinh\sqrt{\sqrt{4}}\big)\%\big) - 4 \times 4$ Steve Wilson, 6/25 Lawrence, KS |
4890 (3.8) $\dfrac{\sqrt{4} - 44\pmf}{4\%\%}$ Steve Wilson, 6/25 Lawrence, KS |
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| 4900 (3.8) $\dfrac{4 - \sqrt{4} - 4\%}{4\%\%}$ Steve Wilson, 6/25 Lawrence, KS |
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| 4910 (4.0) $\dfrac{\sqrt{4} - (4 - .4)\%}{4\%\%}$ Steve Wilson, 6/25 Lawrence, KS |
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| 4920 (4.0) $\dfrac{\sqrt{4} - (\sqrt[.4]{4})\pmf}{4\%\%}$ Steve Wilson, 6/25 Lawrence, KS |
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| 4930 (4.0) $\dfrac{\sqrt{4} - (4! + 4)\pmf}{4\%\%}$ Steve Wilson, 6/25 Lawrence, KS |
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| 4940 (4.0) $\dfrac{4 - \sqrt{4} - 4!\pmf}{4\%\%}$ Steve Wilson, 1/25 Lawrence, KS |
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| 4950 (2.6) $\dfrac{4 - 4\%}{(4 + 4)\%\%}$ Steve Wilson, 6/24 Lawrence, KS |
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| 4956 (3.6) $\dfrac{\sqrt{4}}{4\%\%} - 44$ Steve Wilson, 6/25 Lawrence, KS |
4960 (3.8) $\dfrac{\sqrt{4} - 4 \times 4\pmf}{4\%\%}$ Steve Wilson, 1/25 Lawrence, KS |
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| 4970 (4.2) $\dfrac{\sqrt{4} - \dfrac{4!\pmf}{\sqrt{4}}}{4\%\%}$ Steve Wilson, 1/25 Lawrence, KS |
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| 4975 (3.8) $\dfrac{4 - (\sqrt{4})\%}{(4 + 4)\%\%}$ Steve Wilson, 6/24 Lawrence, KS |
4980 (3.8) $\dfrac{\sqrt{4} - (4 + 4)\pmf}{4\%\%}$ Steve Wilson, 1/25 Lawrence, KS |
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| 4984 (3.6) $\dfrac{\sqrt{4}}{4\%\%} - 4 \times 4$ Steve Wilson, 6/25 Lawrence, KS |
4990 (3.8) $\dfrac{\sqrt{4}}{4\%\%} - \dfrac{4}{.4}$ Steve Wilson, 1/25 Lawrence, KS |
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| 4992 (3.6) $\dfrac{\sqrt{4}}{4\%\%} - 4 - 4$ Steve Wilson, 6/25 Lawrence, KS |
4994 (3.8) $\dfrac{\sqrt{4}}{4\%\%} - \dfrac{4!}{4}$ Steve Wilson, 6/25 Lawrence, KS |
4995 (2.8) $\dfrac{4 - .4\%}{(4 + 4)\%\%}$ Steve Wilson, 6/24 Lawrence, KS |
4996 (2.4) $\dfrac{4}{(4 + 4)\%\%} - 4$ Steve Wilson, 6/24 Lawrence, KS |
4997 (4.2) $\dfrac{\sqrt{4}}{4\%\%} - \sqrt{\dfrac{4}{.\overline{4}}}$ Steve Wilson, 6/25 Lawrence, KS |
4998 (3.8) $\dfrac{\sqrt{4}}{4\%\%} - \dfrac{4}{\sqrt{4}}$ Steve Wilson, 6/25 Lawrence, KS |
4999 (3.6) $\dfrac{\sqrt{4}}{4\%\%} - \dfrac44$ Steve Wilson, 6/25 Lawrence, KS |
5000 (2.4) $\dfrac{4 + 4}{4\% \times 4\%}$ Steve Wilson, 6/24 Lawrence, KS |
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| 5001 (3.6) $\dfrac{\sqrt{4}}{4\%\%} + \dfrac44$ Steve Wilson, 6/25 Lawrence, KS |
5002 (3.8) $\dfrac{\sqrt{4}}{4\%\%} + \dfrac{4}{\sqrt{4}}$ Steve Wilson, 6/25 Lawrence, KS |
5003 (4.2) $\dfrac{\sqrt{4}}{4\%\%} + \sqrt{\dfrac{4}{.\overline{4}}}$ Steve Wilson, 6/25 Lawrence, KS |
5004 (2.4) $\dfrac{4}{(4 + 4)\%\%} + 4$ Steve Wilson, 6/24 Lawrence, KS |
5005 (2.8) $\dfrac{4 + .4\%}{(4 + 4)\%\%}$ Steve Wilson, 6/24 Lawrence, KS |
5006 (3.8) $\dfrac{\sqrt{4}}{4\%\%} + \dfrac{4!}{4}$ Steve Wilson, 6/25 Lawrence, KS |
5008 (3.6) $\dfrac{\sqrt{4}}{4\%\%} + 4 + 4$ Steve Wilson, 6/25 Lawrence, KS |
5010 (3.8) $\dfrac{\sqrt{4}}{4\%\%} + \dfrac{4}{.4}$ Steve Wilson, 1/25 Lawrence, KS |
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| 5016 (3.6) $\dfrac{\sqrt{4}}{4\%\%} + 4 \times 4$ Steve Wilson, 6/25 Lawrence, KS |
5020 (3.8) $\dfrac{\sqrt{4} + (4 + 4)\pmf}{4\%\%}$ Steve Wilson, 6/25 Lawrence, KS |
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| 5025 (3.8) $\dfrac{4 + (\sqrt{4})\%}{(4 + 4)\%\%}$ Steve Wilson, 6/24 Lawrence, KS |
5030 (4.2) $\dfrac{\sqrt{4} + \dfrac{4!\pmf}{\sqrt{4}}}{4\%\%}$ Steve Wilson, 1/25 Lawrence, KS |
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| 5040 (3.8) $\dfrac{\sqrt{4} + 4 \times 4\pmf}{4\%\%}$ Steve Wilson, 1/25 Lawrence, KS |
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| 5044 (3.6) $\dfrac{\sqrt{4}}{4\%\%} + 44$ Steve Wilson, 6/25 Lawrence, KS |
5050 (2.6) $\dfrac{4 + 4\%}{(4 + 4)\%\%}$ Steve Wilson, 6/24 Lawrence, KS |
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| 5060 (4.0) $\dfrac{4 - \sqrt{4} + 4!\pmf}{4\%\%}$ Steve Wilson, 1/25 Lawrence, KS |
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| 5096 (3.8) $\sqrt{\sqrt{4^{4!}}} + \dfrac{4}{4\pmf}$ Steve Wilson, 1/25 Lawrence, KS |
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| 5116 (5.6) $\dfrac{4^{\sec\arctan\sqrt{4!}}}{\cos\arctan\sqrt{4!}} - 4$ Bruce Manoly, 5/24 Crescent City, CA |
5118 (5.8) $\dfrac{4^{\sec\arctan\sqrt{4!}}}{\cos\arctan\sqrt{4!}} - \sqrt{4}$ Bruce Manoly, 5/24 Crescent City, CA |
5120 (3.4) $\dfrac{4 \times 4^4}{\sqrt{4\%}}$ Steve Wilson, 1/25 Lawrence, KS |
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| 5122 (5.8) $\dfrac{4^{\sec\arctan\sqrt{4!}}}{\cos\arctan\sqrt{4!}} + \sqrt{4}$ Bruce Manoly, 5/24 Crescent City, CA |
5124 (5.6) $\dfrac{4^{\sec\arctan\sqrt{4!}}}{\cos\arctan\sqrt{4!}} + 4$ Bruce Manoly, 5/24 Crescent City, CA |
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| 5184 (3.2) $4 \times \left( \dfrac{4!}{4} \right)^4$ Marc Carreras-Vidal, 8/15 Maó, Spain |
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| 5250 (3.8) $\dfrac{4 + \sqrt{4\%}}{(4 + 4)\%\%}$ Steve Wilson, 6/24 Lawrence, KS |
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| 5280 (5.0) $\dfrac{44 \times 4!}{\cos\arctan\sqrt{4!}}$ Bruce Manoly, 5/24 Crescent City, CA |
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| 5300 (3.8) $\dfrac{4 + 4!\%}{(4 + 4)\%\%}$ Steve Wilson, 6/24 Lawrence, KS |
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| 5500 (2.4) $\dfrac{4.4}{(4 + 4)\%\%}$ Steve Wilson, 6/24 Lawrence, KS |
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| 5625 (2.8) $\dfrac{4}{4\% \times 4\% \times .\overline{4}}$ Steve Wilson, 6/24 Lawrence, KS |
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| 5720 (10.0) $\cot\arccsc(f_{4})$ $\phantom. \times \sec\arctan(4!!)$ $\phantom. \times \cot\arccsc(((f_4)!)!!!!)$ $\phantom. \times \cot\arccsc(f_{4!!})$ Bruce Manoly, 4/25 Crescent City, CA |
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| 5760 (5.4) $\dfrac{(4! + 4!) \times 4!}{\cos\arctan\sqrt{4!}}$ Bruce Manoly, 5/24 Crescent City, CA |
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| 5832 (6.2) $\dfrac{\cosh\ln\sqrt{4\%}}{.\overline{4}\pmf}$ $\phantom. - \dfrac{4!}{\csch\ln\sqrt{4}}$ Steve Wilson, 6/25 Lawrence, KS |
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| 5991 (3.8) $\dfrac{4!}{4\pmf} - \dfrac{4}{.\overline{4}}$ Steve Wilson, 12/24 Lawrence, KS |
5992 (3.4) $\dfrac{4!}{4\pmf} - 4 - 4$ Steve Wilson, 12/24 Lawrence, KS |
5993 (3.8) $\dfrac{4! - (4! + 4)\pmf}{4\pmf}$ Steve Wilson, 12/24 Lawrence, KS |
5994 (3.6) $\dfrac{4!}{4\pmf} - 4 - \sqrt{4}$ Steve Wilson, 12/24 Lawrence, KS |
5995 (3.6) $\dfrac{4! - 4\pmf}{4\pmf} - 4$ Steve Wilson, 12/24 Lawrence, KS |
5996 (3.6) $\dfrac{4!}{4\pmf} - \sqrt{4 \times 4}$ Steve Wilson, 12/24 Lawrence, KS |
5997 (3.6) $\dfrac{4! + 4\pmf}{4\pmf} - 4$ Steve Wilson, 12/24 Lawrence, KS |
5998 (3.6) $\dfrac{4!}{4\pmf} - 4 + \sqrt{4}$ Steve Wilson, 12/24 Lawrence, KS |
5999 (3.4) $\dfrac{4!}{4\pmf} - \dfrac44$ Steve Wilson, 12/24 Lawrence, KS |
6000 (2.6) $\dfrac{4 - 4 \times .4}{4\%\%}$ Steve Wilson, 6/24 Lawrence, KS |
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| 6048 (3.2) $(4^4 - 4) \times 4!$ Steve Wilson, 11/24 Lawrence, KS |
6050 (3.8) $\dfrac{4! + \dfrac{.4}{\sqrt{4}}}{4\pmf}$ Steve Wilson, 1/25 Lawrence, KS |
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| 6120 (3.4) $4^4 \times 4! - 4!$ Josh Crow, 1/09 Lenexa, KS |
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6140 (3.2) $4^4 \times 4! - 4$ Jeremy Miller, 8/07 Olathe, KS |
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| 6142 (3.4) $4^4 \times 4! - \sqrt{4}$ Steve Wilson, 6/24 Lawrence, KS |
6144 (8.8) $\sqrt{\sqrt{(\sqrt{4} + \sqrt{-4})^{4!}}}$ $\phantom. \times ((f_4)!)!!!!$ Bruce Manoly, 4/25 Crescent City, CA |
6146 (3.4) $4^4 \times 4! + \sqrt{4}$ Steve Wilson, 6/24 Lawrence, KS |
6148 (3.2) $4^4 \times 4! + 4$ Jeremy Miller, 8/07 Olathe, KS |
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| 6168 (3.4) $4^4 \times 4! + 4!$ Josh Crow, 1/09 Lenexa, KS |
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| 6250 (2.6) $\dfrac{4}{4\% \times 4\% \times .4}$ Steve Wilson, 6/24 Lawrence, KS |
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| 6311 (7.2) $(4!!)!!! \times (4!!)!!! - (4!!)!!!$ $\phantom. - \sec\arctan\sqrt{(4!!)!!!}$ Bruce Manoly, 4/25 Crescent City, CA |
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| 6660 (3.8) $\dfrac{444}{\sqrt{.\overline{4}\%}}$ David Barksdale, 3/09 Kirkland, WA |
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| 6912 (3.6) $\dfrac{4!^4}{4! + 4!}$ Bruce Manoly, 5/24 Crescent City, CA |
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| 7200 (2.6) $(4 + 4) \times \dfrac{4}{.\overline{4}\%}$ Steve Wilson, 6/24 Lawrence, KS |
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| 7500 (2.4) $\dfrac{4 - \dfrac44}{4\%\%}$ Steve Wilson, 6/24 Lawrence, KS |
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| 7744 (3.2) $4 \times \sqrt{44^4}$ Regina Hillman, 10/07 Bucyrus, KS |
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| 7776 (3.6) $\left(\dfrac{4!}{4}\right)^{\sqrt{4}/.4}$ Steve Wilson, 6/24 Lawrence, KS |
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| 8000 (2.4) $(4 + 4) \times \dfrac{4}{.4\%}$ Steve Wilson, 6/24 Lawrence, KS |
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| 8701 (5.2) $\dfrac{4! \times \cosh\ln(.4) + 4\pmf}{4\pmf}$ Steve Wilson, 11/24 Lawrence, KS |
8702 (5.2) $\dfrac{4! \times \cosh\ln(.4)}{4\pmf} + \sqrt{4}$ Steve Wilson, 11/24 Lawrence, KS |
8704 (5.0) $\dfrac{4! \times \cosh\ln(.4)}{4\pmf} + 4$ Steve Wilson, 11/24 Lawrence, KS |
8706 (5.4) $\bigg(\dfrac{\cosh\ln(.4)}{4\pmf}$ $\phantom. + \cot\arctan 4\bigg) \times 4!$ Steve Wilson, 11/24 Lawrence, KS |
8707 (6.0) $\coth\ln\coth\arcosh\sqrt{4}$ $\phantom. + \dfrac{4! \times \cosh\ln(.4)}{4\pmf}$ Steve Wilson, 11/24 Lawrence, KS |
8708 (5.8) $\bigg(\dfrac{\cosh\ln(.4)}{4\pmf}$ $\phantom. + \tanh\ln\sqrt{\sqrt{4}}\bigg) \times 4!$ Steve Wilson, 11/24 Lawrence, KS |
8710 (5.2) $\dfrac{4! \times \cosh\ln(.4) + 4\%}{4\pmf}$ Steve Wilson, 11/24 Lawrence, KS |
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| 8748 (6.0) $\cosh(\Gamma(4) \times \arsinh\sqrt{\Gamma(4)})$ $\phantom. - \dfrac44$ Steve Wilson, 6/25 Lawrence, KS |
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| 8800 (3.4) $\dfrac{44 \times 4}{(\sqrt{4})\%}$ Steve Wilson, 6/24 Lawrence, KS |
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| 9000 (2.6) $\dfrac{44 - 4}{.\overline{4}\%}$ David Barksdale, 3/09 Kirkland, WA |
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| 9216 (3.4) $4! \times 4! \times 4 \times 4$ Steve Wilson, 11/24 Lawrence, KS |
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| 10000 (3.2) $\left( \dfrac{4!}{4} + 4\right)^4$ Steve Wilson, 4/24 Lawrence, KS |
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| 10368 (4.0) $\left(\dfrac{4!}{4}\right)!^{\sqrt{4}} \times (\sqrt{4})\%$ Steve Wilson, 11/24 Lawrence, KS |
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| 11100 (2.2) $\dfrac{444}{4\%}$ Dave Jones, 9/06 Coventry, England |
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| 11250 (3.2) $\dfrac{4}{\left(.\overline{4} + .\overline{4}\right) \times 4\%\%}$ Steve Wilson, 11/24 Lawrence, KS |
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| 11264 (3.0) $44 \times 4^4$ Joseph Mavungu, 5/03 Olathe, KS |
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| 12288 (3.4) $4^4 \times 4! \times \sqrt{4}$ Jeremy Miller, 8/07 Olathe, KS |
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| 13800 (3.8) $4! \times 4! \times 4! - 4!$ Zach Warring, 9/09 Olathe, KS |
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| 13820 (3.6) $4! \times 4! \times 4! - 4$ Jeremy Miller, 9/07 Olathe, KS |
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| 13824 (3.2) $(4!)^{4 - 4/4}$ Steve Wilson, 11/24 Lawrence, KS |
13828 (3.6) $4! \times 4! \times 4! + 4$ Jeremy Miller, 9/07 Olathe, KS |
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| 13848 (3.8) $4! \times 4! \times 4! + 4!$ Zach Warring, 9/09 Olathe, KS |
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| 14641 (3.0) $\left( \dfrac{44}{4} \right)^4$ Brian Nixt, 3/06 Lenexa, KS |
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| 15000 (3.4) $4! \times \left( \dfrac{\sqrt{4}}{.4} \right)^4$ Bruce Manoly, 5/24 Crescent City, CA |
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| 15127 (8.2) $\dfrac{f_{(4!!)!!!! + 4!!}}{f_{4! - 4}}$ Bruce Manoly, 4/25 Crescent City, CA |
15128 (7.2) $\dfrac{f_{4! + (f_4)!}}{f_{4/.4}}$ Bruce Manoly, 4/25 Crescent City, CA |
Page 1 (1-400), Page 2 (401-800), Page 3 (801-1200), Page 4 (1201-1600), Page 5 (1601-2000), Page 6 (2001-2400), Page 7 (2401-2800), Page 8 (2801-3200), Page 9 (3201-3600), Page 10 (3601-4000), Page 11 (4001-4400), Page 12 (4401-4800), Page 13 (4801-16000), Page 41 (16001+).