Express 6
I was given this puzzle at the last Gathering for Gardner.
Use arithmetic operations to express 6 using three identical digits. For each digit from 0 to 9 find at least one way to express 6.
For example, I can express 6 using three twos in many ways: 2 + 2 + 2, or 2 · 2 + 2, or 22 + 2. But the problem doesn’t ask for many ways. One way is enough, but you need to do it for every digit. So nine more cases to go: 0, 1, 3, 4, 5, 6, 7, 8, and 9.
Share:
George R.:
Well, I’ll try the easy ones first 🙂
22 May 2014, 11:43 am6*(6/6)
5+5/5
7-7/7
sqrt4 +sqrt4 +sqrt4
3*3 -3
sqrt (9*9)-sqrt9
(0! +0!+0!)!
(1+1+1)!
I don’t have a solution for 8’s yet.
Tanya Khovanova:
George, that was fast. One more to go.
22 May 2014, 12:20 pmKonstantin Knop:
8 – sqrt(sqrt(8+8))
22 May 2014, 1:04 pmTanya Khovanova:
Konstantin,
Nice, I know a different one.
22 May 2014, 3:41 pmLeo:
log(cbrt(8), 8*8)
22 May 2014, 7:01 pmLeo:
OK, now seriously:
log(sqrt(8)*8)/log(sqrt(sqrt(8)))
22 May 2014, 7:08 pmLeo:
This works for all digits except 0 and 1, too!
22 May 2014, 7:11 pmtanyakh:
The one I knew for 8: (sqrt(8+8/8))!
12 February 2026, 12:19 pmtanyakh:
Also, 8!!/(8*8).
12 February 2026, 1:02 pmtanyakh:
Also for 8: ceiling(8*.8-.8).
28 February 2026, 3:33 pmtanyakh:
Also for 9: floor(.9) + sqrt(9) + sqrt(9).
28 February 2026, 3:34 pmtanyakh:
For 8: ceiling((8+8)/ceiling(sqrt(8)));
2 March 2026, 2:30 pm8!!/sqrt(8*8);
ceiling(sqrt(8))! + 8 – 8;
ceiling(sqrt(8))*ceiling(sqrt(8))-ceiling(sqrt(8)),
For 9: sqrt(sqrt(9))sqrt(sqrt(9)) + sqrt(9);
9 – 9/aqrt(9);
sqrt(9)! + 9 – 9.