Category Archives: Arithmetic


In this post I’d like to introduce TooTs, a mix between crossword puzzles and numbers. The grid looks just like a crossword puzzle, but instead of words the grid has to be filled with numbers. Vertical numbers must be read top-down. Thus if the digits 3, 9 and 5 are listed from the top down, the number would be 395.

Every clue consists of three numbers. Two of them have to be added together to get the number to be filled into the grid.
Example: the clue is 7, 8 and 13. Then the solution is either 7+8=15, 7+13=20 or 8+13=21. The name TooT is shorthand for Two out of Three.

Here is a 5×5 exercise:
Toot 5x5 2015-04-24 exercise

1) 16, 17, 18
3) 20, 26, 36
4) 142, 139, 145
8) 6819, 20002, 30134
11) 18, 20, 22
12) 11, 24, 36
2) 17, 19, 23
3) 18, 36, 47
5) 400, 406, 418
6) 18, 106, 256
7) 15, 25, 190
9) 1, 51, 61
10) 11, 12, 13

A 7×7 exercise:
Toot 7x7 2015-04-24 exercise

1) 16891 18930
6) 382, 23, 67
8) 25, 8, 17
10) 32, 14, 17
11) 2913476, 173823, 1876543
12) 61, 23, 38
13) 45, 11, 34
14) 865, 249, 444
16) 13947, 1171, 5419
2) 53, 26, 27
3) 8843269, 332160, 345612
4) 22, 3, 5
5) 12263, 5321, 6942
7) 62652, 23487, 39165
9) 591, 109, 482
10) 374, 25, 98
14) 83, 16, 26
15) 54, 17, 27

You can check your solution here and here

A 9×9 puzzle:
Toots 9x9 2015-05-15 nr 1

1. 108, 132, 146
4. 2, 166, 660
6. 2497, 9892, 12837
9. 0, 7, 24
11. 212, 669, 774
12. 4, 19, 30
13. 18, 27, 27
15. 14, 33, 40
16. 242, 977, 2236
17. 596, 903, 2770
18. 25, 31, 52
20. 4, 11, 22
21. 7, 9, 35
22. 126, 343, 422
24. 3, 10, 13
26. 2918, 74181, 82214
28. 292, 320, 398
29. 66, 191, 228
1. 38, 96, 224
2. 4, 41, 77
3. 239, 1644, 4146
4. 19, 29, 35
5. 3, 7, 227
7. 20, 36, 38
8. 1, 14, 17
10. 12591, 13966, 31881
12. 706, 10961, 36955
14. 186, 210, 367
15. 102, 153, 279
19. 2287, 3330, 3945
21. 112, 239, 304
22. 19, 26, 45
23. 6, 23, 87
25. 74, 299, 315
26. 33, 49, 52
27. 12, 12, 12

You can check your solution here and here

In a subsequent post, probably next month, I hope to publish some variations.



A man is 25 years old and his wife 23. He noticed that the sum of their ages (25+23=48) is exactly 4 times the sum of the digits of their ages. (2+5+2+3=12).

When will the sum of their ages be exactly 8 times the sum of the digits of their ages? And when will it be 9 times the sum of the digits?

You can check your solutions here

New puzzles are published at least twice a month on Friday. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.


The Dutch reformed-christian ‘Reformatorisch Dagblad’ twice a year publishes an extra puzzle issue for its subscribers. This weeks puzzle type is 1-8, invented by Marijke Balmaekers, and published in the childrens section of the ‘Vakantie Doe Boek’ of the reformatorisch Dagblad.

The numbers one to eight have been arranged in a 5×5 grid in such a way that:

  1. Each of the numbers one to eight is used exactly once

  2. There are always one or two numbers in every row, column or diagonal

  3. the sum of the numbers is listed as a clue at the end of the row/column/diagonal

1-8 example

number 1
1-8 2016-02-14 nr 1 exercise

number 2
1-8 2016-02-14 nr 2 exercise

number 3
1-8 2016-02-14 nr 3 exercise

You can check your solutions here, here and here

The online exam

certificate illustration
This week I’ve got a quickie for you.
Last week I took an online certification exam. It was an open book certifiction, and I was free to consult the website and course map as often and as long as I wanted. Some types of questions scored 3 points, others scored 5 points.

My result was:
You scored 201 points out of 223 total possible points.
You answered 45 out of 51 questions correctly.

How many 5-point questions and how many 3 point questions did I miss?

You can check your solutions here

A new puzzle is posted every friday. You are welcome to comment on the puzzles. Solutions are added at the bottom of a puzzle after one or more weeks.

A couple of oldies

1) The windmills
The Jones family was en route to their holiday home. ‘Look Johnny,’ Dad said, ‘There is a row of 4 windmills, all evenly paced apart’. Johnny took his Dads smart phone and activated the stopwatch as they passed the first one.
‘Look Dad,’ he said. It took us 10 seconds to pass these 4 windmills!
‘Well done, son’ his Dad complimented him. ‘Now there is a row of 7 windmills coming up. They seem to be the same type, so they will have the same distance between them. How long will it take us to pass them?’

You can check your solutions here

This is one of what I call a class of puzzles, of which over the ages there has developed a large number of variations. The puzzle above is often presented with trees along a road. Here is another variation which I came across several times:

2) The clock
The tower clock at Backditch is rightly famous for both its artwork decorations and its accuracy. Earlier this week we were able to visit it and the church warden was willing to show us the centuries old woodwork driving the clock.
Just when we were on top the clock struck 6 o clock and I noticed that it took exactly 12 seconds.
The warden asked me: How long will it take to strike 11 o clock?

You can check your solutions here

The commons element in these puzzles is of course that the important item in not the moments, but the intervals in between. A variation can be found in nearly every puzzle collection.

Now many of the puzzles we have seen around for a long time were first published by someone called Alquin of York in a booklet called Propositiones ad acuendos iuvenes (“Problems to sharpen the young”), probably written around 800AD. This one, however, was not. There is however a related problem, which is totally unknown, that does appear in this antique puzzle collection:

3) The farmer
How many furrows might a farmer have in his field if the ploughman makes three turns at each end of the field?

You can check your solutions here
(solve the puzzle above before reading on!)

(solve the puzzle above before reading on!)

(solve the puzzle above before reading on!)

(solve the puzzle above before reading on!)
The important elements in Alquins puzzle are:
– it’s not the turns, but the furrows in between the turns that count
– the first or last furrow, depending on the view, does not need a turn.

That the end deserves a special treatment, is also the subject of another puzzle:
4) The snail
A snake is at the bottom of a pit with 3 meter high walls. Every day the snail climbs 1 meter, but when he sleeps he slides back 60 cm. How many days does he need to crawl out?

You can check your solutions here

5) The Adventurous Snail
In his “The Canterbury puzzles”, Henry Dudeney publishes a small variation:
adventurous snail 152

A simple version of the puzzle of the climbing snail is familiar to everybody. We were all taught it in the nursery, and it was apparently intended to inculcate the simple moral that we should never slip if we can help it. This is the popular story. A snail crawls up a pole 12 feet high, ascending 3 feet every day and slipping back 2 feet every night. How long does it take to get to the top? Of course, we are expected to say the answer is twelve days, because the creature makes an actual advance of 1 foot in every twenty-four hours. But the modern infant in arms is not taken in in this way. He says, correctly enough, that at the end of the[Pg 153] ninth day the snail is 3 feet from the top, and therefore reaches the summit of its ambition on the tenth day, for it would cease to slip when it had got to the top.

Let us, however, consider the original story. Once upon a time two philosophers were walking in their garden, when one of them espied a highly respectable member of the Helix Aspersa family, a pioneer in mountaineering, in the act of making the perilous ascent of a wall 20 feet high. Judging by the trail, the gentleman calculated that the snail ascended 3 feet each day, sleeping and slipping back 2 feet every night.

“Pray tell me,” said the philosopher to his friend, who was in the same line of business, “how long will it take Sir Snail to climb to the top of the wall and descend the other side? The top of the wall, as you know, has a sharp edge, so that when he gets there he will instantly begin to descend, putting precisely the same exertion into his daily climbing down as he did in his climbing up, and sleeping and slipping at night as before.”

This is the true version of the puzzle, and my readers will perhaps be interested in working out the exact number of days. Of course, in a puzzle of this kind the day is always supposed to be equally divided into twelve hours’ daytime and twelve hours’ night.

You can check your solutions here

6) The beggar and the cigarettes
A beggar needs 7 cigarette buts to make one new one. After digging through several garbage cans he collected 55 cigarette buds. How many cigarettes can he smoke?

You can check your solutions here