Eight eights that add to a thousand

Question

Using exactly eight 8s and nothing but plus signs, write an expression equal to 1000. You may join 8s together to form numbers like 88 or 888. Then find a way to make 1000 with eight 8s using other operations, and explain why 1000 cannot be made from eight 8s with plus signs in any essentially different way.

Accepted Answer

888 + 88 + 8 + 8 + 8 = 1000. Check. 888 + 88 = 976; 976 + 8 + 8 + 8 = 1000. Eight 8s used: three in 888, two in 88, three single 8s. Why this is the only addition solution. The number 1000 ends in 0, and every term ends in 8, so the units digits of the terms must add to a multiple of 10: with k terms that is 8k, and 8k ends in 0 only for k = 5 (40) or k = 10 (80). Only five terms are possible with eight 8s (ten terms would need ten 8s). So there are exactly five terms whose units digits contribute 40, carrying 4 into the tens. The tens digits must then add with the carry to something ending i…

Eight eights that add to a thousand

Your answer

Discussion

Learn the concepts

Related questions