Â鶹´«Ã½

#84 Wild ones

Set by Alison Kiddle

Using addition and multiplication, and brackets, we can repeatedly use 1 to make the number 12: (1 + 1) × (1 + 1 + 1 + 1 + 1 + 1). This uses eight 1s.

Can you find a way to make the number 12 using fewer than eight 1s?

What is the largest number you can make using exactly nine 1s?

What is the smallest number of 1s you need to make the number 51?

Solution next week

#83 Doubled squares

Solution

There are many numbers that fit the bill. For instance, 289 is 17² and 1 more than 2 × 12².

A four-digit example is 9801, which is 99² and 1 more than 2 × 70².

The smallest pair of these squares where each has a different number of digits is 11,309,769 (3363², eight digits) and 5,654,884 (2378², seven digits).

Numbers n such that n² = 2m² + 1 form a sequence where each value is six times the previous value, minus the value before that. The sequence starts 1, 3, 17, 99…