Puzzles
23 December
153 is equal to the sum of the cubes of its digits: 13 + 53 + 33.
There are three other three-digit numbers that are equal to the sum of the cubes of their digits. What is the largest of these numbers?
19 December
Eve uses the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 to write five square numbers (using each digit exactly once). What is largest square number that she made?
12 December
Mary uses the digits 1, 2, 3, 4, 5, 6 and 7 to make two three-digit numbers and a one-digit number (using each digit exactly once). The sum of her three numbers is 1000.
What is the smallest that the larger of her two three-digit numbers could be?
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The sum of the units digits of the three numbers must be 10 (as the units digit of the result must be 0 and the number are not big enough to allow the sum to be 20). Similarly, the sum of the
two tens digits must be 9, and the sum of the two hundreds digits is 9 (so that 10s are made when the 1 is carried).
The> hundreds digits could be 2 and 7, 3 and 6, or 4 and 5. As we're looking to make the larger number as small as possible, we should pick 4 and 5.
The tens digits could be 2 and 7 or 3 and 6. We pick 2 and 7 and as 2 is the smallest possible number here.
Finally, the units digits are 1, 3 and 6. We use 1 for our number as it's smallest, leading to 521.
5 December
The number 36 is equal to two times the product of its digits.
What is the only (strictly positive) number that is equal to four times the product of its digits?
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We know the answer has three digits: we are looking for a number ABC that is equal to 4×A×B×C. Dividing by A, we see that 4×B×C > 100 and so B×C > 25. This
means that the only possible options for BC are 65, 66, 74, 75, 76, 77, 84, 85, 86, 87, 88, 93, 94, 95, 96, 97, 98 and 99.
Trying these options with different values for A, we find that 384 works.
3 December
Holly picks the number 513, reverses it to get 315, then adds the two together to make 828.
Ivy picks a three-digit number, reverses it, then adds the two together to make 968. What is the smallest number that Ivy could have started with?
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If Ivy's number is ABC, then A+C must be 8 or 18 (as this makes this final digit of the sum correct). A+C cannot be 18 of the sum would be bigger than 1000, so A+C must be 8.
The middle digit tells us that 2×B must be 6 or 16. As the first digit of the sum is 9 and not 8, there must be a 1 carried over into the hundreds column, so 2×B = 16 and B = 8.
Overall, we know that Ivy's number is A8C, with A+C = 8. The smallest possible number Ivy could have started with is therefore 187.
2 December
Eve writes down the numbers from 1 to 10 (inclusive). In total she write down 11 digits.
Noel writes down the number from 1 to 100 (inclusive). How many digits does he write down?
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Noel writes down 9 one-digit numbers, 90 two-digit numbers and 1 three-digit number. In total, these include 1×9 + 2×90 + 3×1 = 192 digits.
1 December
Some numbers contain a digit more than once (eg 313, 111, and 144). Other numbers have digits that are all different (eg 123, 307, and 149).
How many three-digit numbers are there whose digits are all different?
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There are 9 options for the first digit of the number (1 to 9). There are then 9 options for the second digit (0 to 9 excluding the digit already used),
and 8 options for the third digit (0 to 9 excluding the two digits already used). In total, this makes 9×9×8 = 648 numbers whose digits are all different.
24 December
There are 343 three-digit numbers whose digits are all 1, 2, 3, 4, 5, 6, or 7. What is the
mean of all these numbers?
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The digits 1 to 7 will each appear the same number of times as each other in each position of the number, so each digit of the mean will be the mean of the digits 1 to 7.
Therefore the mean is 444.
